![6 SOME BACKGROUND ON PROBABILITY
which justifies (1.2).
If the probability of B is unaffected by the prior occurrence of 𝐴, then we say that
𝐴 and 𝐵 are independent or that
P(𝐵∣𝐴) = P(𝐵),
which from above implies that
P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵).
Conversely, if P(𝐵∣𝐴) = P(𝐵), then 𝐴 and 𝐵 are independent events. Again this
result can be extended to 3 or more independent events.
Example 1.3 Let 𝐴 and 𝐵 be independent events with P(𝐴) = 1
4
and P(𝐵) = 2
3
. Calculate
the following probabilities: (a) P(𝐴 ∩ 𝐵); (b) P(𝐴 ∩ 𝐵𝑐
); (c) P(𝐴𝑐
∩ 𝐵𝑐
); (d) P(𝐴𝑐
∩ 𝐵);
(e) P((𝐴 ∪ 𝐵)𝑐
).
Since the events are independent, then P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵). Hence
(a) P(𝐴 ∩ 𝐵) = 1
4
⋅ 2
3
= 1
6
.
(b) The independence 𝐴 and 𝐵𝑐
follows by eliminating P(𝐴 ∩ 𝐵) between the equations
P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵) = P(𝐴)[1 − P(𝐵𝑐
)]
and
P(𝐴) = P[(𝐴 ∩ 𝐵𝑐
) ∪ (𝐴 ∩ 𝐵)] = P(𝐴 ∩ 𝐵𝑐
) + P(𝐴 ∩ 𝐵).
Hence
P(𝐴 ∩ 𝐵𝑐
) = P(𝐴)P(𝐵𝑐
) = P(𝐴)[1 − P(𝐵)] = 1
4
(1 − 2
3
) = 1
12
.
(c) Since 𝐴𝑐
and 𝐵𝑐
are independent events,
P(𝐴𝑐
∩ 𝐵𝑐
) = P(𝐴𝑐
)P(𝐵𝑐
) = [1 − P(𝐴)][1 − P(𝐵)] = 3
4
⋅ 1
3
= 1
4
.
(d) Since 𝐴𝑐
and 𝐵 are independent events, P(𝐴𝑐
∩ 𝐵) = P(𝐴𝑐
)P(𝐵) = [1 − 1
4
]2
3
= 1
2
.
(e) P((𝐴 ∪ 𝐵)𝑐
) = 1 − P(𝐴 ∪ 𝐵) = 1 − P(𝐴) − P(𝐵) + P(𝐴 ∩ 𝐵) by (1.1). Hence
P((𝐴 ∪ 𝐵)𝑐
) = 1 − P(𝐴) − P(𝐵) + P(𝐴)P(𝐵) = 1 − 1
4
− 2
3
+ 1
6
= 1
4
.
Example 1.4. For three events 𝐴, 𝐵, and 𝐶, show that
P(𝐴 ∩ 𝐵∣𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵∣𝐶),
where P(𝐶) > 0.
By using (1.2) and viewing 𝐴 ∩ 𝐵 ∩ 𝐶 as (𝐴 ∩ 𝐵) ∩ 𝐶 or 𝐴 ∩ (𝐵 ∩ 𝐶),
P(𝐴 ∩ 𝐵 ∩ 𝐶) = P(𝐴 ∩ 𝐵∣𝐶)P(𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵 ∩ 𝐶).
Hence
P(𝐴 ∩ 𝐵∣𝐶) = P(𝐴∣𝐵 ∩ 𝐶)
P(𝐵 ∩ 𝐶)
P(𝐶)
= P(𝐴∣𝐵 ∩ 𝐶)P(𝐵∣𝐶)
by (1.2) again.
A result known as the law of total probability or the partition theorem will](https://image.slidesharecdn.com/63744-250715022432-b89e2119/85/Stochastic-Processes-An-Introduction-2nd-Edition-Peter-Watts-Jones-22-320.jpg)
Stochastic Processes An Introduction 2nd Edition Peter Watts Jones
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- 11. Contents Preface ix 1 Some Background on Probability 1 1.1 Introduction 1 1.2 Probability 1 1.3 Conditional probability and independence 5 1.4 Discrete random variables 7 1.5 Continuous random variables 9 1.6 Mean and variance 10 1.7 Some standard discrete probability distributions 12 1.8 Some standard continuous probability distributions 14 1.9 Generating functions 17 1.10 Conditional expectation 21 1.11 Problems 24 2 Some Gambling Problems 29 2.1 Gambler’s ruin 29 2.2 Probability of ruin 29 2.3 Some numerical simulations 33 2.4 Duration of the game 34 2.5 Some variations of gambler’s ruin 38 2.5.1 The infinitely rich opponent 38 2.5.2 The generous opponent 38 2.5.3 Changing the stakes 39 2.6 Problems 39 3 Random Walks 45 3.1 Introduction 45 3.2 Unrestricted random walks 46 3.3 The general probability distribution of a walk 48 3.4 First returns of the symmetric random walk 50 3.5 Problems 52 4 Markov Chains 59 4.1 States and transitions 59 4.2 Transition probabilities 60 v
- 12. vi CONTENTS 4.3 General two-state Markov chains 64 4.4 Powers of the general transition matrix 66 4.5 Gambler’s ruin as a Markov chain 73 4.6 Classification of states 76 4.7 Classification of chains 83 4.8 Problems 86 5 Poisson Processes 93 5.1 Introduction 93 5.2 The Poisson process 93 5.3 Partition theorem approach 96 5.4 Iterative method 97 5.5 The generating function 98 5.6 Variance in terms of the probability generating function 100 5.7 Arrival times 101 5.8 Summary of the Poisson process 103 5.9 Problems 104 6 Birth and Death Processes 107 6.1 Introduction 107 6.2 The birth process 107 6.3 Birth process: Generating function equation 110 6.4 The death process 112 6.5 The combined birth and death process 115 6.6 General population processes 119 6.7 Problems 122 7 Queues 131 7.1 Introduction 131 7.2 The single-server queue 132 7.3 The stationary process 134 7.4 Queues with multiple servers 140 7.5 Queues with fixed service times 144 7.6 Classification of queues 147 7.7 A general approach to the 𝑀(𝜆)/𝐺/1 queue 147 7.8 Problems 151 8 Reliability and Renewal 157 8.1 Introduction 157 8.2 The reliability function 157 8.3 Exponential distribution and reliability 159 8.4 Mean time to failure 160 8.5 Reliability of series and parallel systems 161 8.6 Renewal processes 163 8.7 Expected number of renewals 165
- 13. CONTENTS vii 8.8 Problems 167 9 Branching and Other Random Processes 171 9.1 Introduction 171 9.2 Generational growth 171 9.3 Mean and variance 174 9.4 Probability of extinction 176 9.5 Branching processes and martingales 179 9.6 Stopping rules 182 9.7 The simple epidemic 184 9.8 An iterative solution scheme for the simple epidemic 186 9.9 Problems 188 10 Computer Simulations and Projects 195 Answers and Comments on End-of-Chapter Problems 203 Appendix 211 References and Further Reading 215 Index 217
- 15. Preface This textbook was developed from a course in stochastic processes given by the au- thors over many years to second-year students studying Mathematics or Statistics at Keele University. At Keele the majority of students take degrees in Mathematics or Statistics jointly with another subject, which may be from the sciences, social sci- ences or humanities. For this reason the course has been constructed to appeal to students with varied academic interests, and this is reflected in the book by including applications and examples that students can quickly understand and relate to. In par- ticular, in the earlier chapters, the classical gambler’s ruin problem and its variants are modeled in a number of ways to illustrate simple random processes. Specialized applications have been avoided to accord with our view that students have enough to contend with in the mathematics required in stochastic processes. Topics can be selected from Chapters 2 to 9 for a one-semester course or mod- ule in random processes. It is assumed that readers have already encountered the usual first-year courses in calculus and matrix algebra and have taken a first course in probability; nevertheless, a revision of relevant basic probability is included for reference in Chapter 1. Some of the easier material on discrete random processes is included in Chapters 2, 3, and 4, which cover some simple gambling problems, ran- dom walks, and Markov chains. Random processes continuous in time are developed in Chapters 5 and 6. These include Poisson, birth and death processes, and general population models. Continuous time models include queues in Chapter 7, which has an extended discussion on the analysis of associated stationary processes. The book ends with two chapters on reliability and other random processes, the latter including branching processes, martingales, and a simple epidemic. An appendix contains key mathematical results for reference. There are over 50 worked examples in the text and 205 end-of-chapter problems with hints and answers listed at the end of the book. Mathematica𝑇 𝑀 is a mathematical software package able to carry out complex symbolic mathematical as well as numerical computations. It has become an integral part of many degree courses in Mathematics or Statistics. The software has been used throughout the book to solve both theoretical and numerical examples and to produce many of the graphs. R is a statistical computing and graphics package which is available free of charge, and can be downloaded from: http://www.r-project.org ix
- 16. x PREFACE Like R, S-PLUS (not freeware) is derived from the S language, and hence users of these packages will be able to apply them to the solution of numerical projects, including those involving matrix algebra presented in the text. Mathematica code has been applied to all the projects listed by chapters in Chapter 10, and R code to some as appropriate. All the Mathematica and R programs can be found on the Keele University Web site: http://www.scm.keele.ac.uk/books/stochastic processes/ Not every topic in the book is included, but the programs, which generally use standard commands, are intended to be flexible in that inputs, parameters, data, etc., can be varied by the user. Graphs and computations can often add insight into what might otherwise be viewed as rather mechanical analysis. In addition, more compli- cated examples, which might be beyond hand calculations, can be attempted. We are grateful to staff of the School of Computing and Mathematics, Keele Uni- versity, for help in designing the associated Web site. Finally, we would like to thank the many students at Keele over many years who have helped to develop this book, and to the interest shown by users of the first edition in helping us to refine and update this second edition. Peter W. Jones Peter Smith Keele University
- 17. CHAPTER 1 Some Background on Probability 1.1 Introduction We shall be concerned with the modeling and analysis of random experiments us- ing the theory of probability. The outcome of such an experiment is the result of a stochastic or random process. In particular we shall be interested in the way in which the results or outcomes vary or evolve over time. An experiment or trial is any sit- uation where an outcome is observed. In many of the applications considered, these outcomes will be numerical, sometimes in the form of counts or enumerations. The experiment is random if the outcome is not predictable or is uncertain. At first we are going to be concerned with simple mechanisms for creating random outcomes, namely games of chance. One recurring theme initially will be the study of the classical problem known as gambler’s ruin. We will then move on to applications of probability to modeling in, for example, engineering, medicine, and biology. We make the assumption that the reader is familiar with the basic theory of probability. This background will however be reinforced by the brief review of these concepts which will form the main part of this chapter. 1.2 Probability In random experiments, the list of all possible outcomes is termed the sample space, denoted by 𝑆. This list consists of individual outcomes or elements. These elements have the properties that they are mutually exclusive and that they are exhaustive. Mutually exclusive means that two or more outcomes cannot occur simultaneously: exhaustive means that all possible outcomes are in the list. Thus each time the exper- iment is carried out one of the outcomes in 𝑆 must occur. A collection of elements of 𝑆 is called an event: these are usually denoted by capital letters, 𝐴, 𝐵, etc. We denote by P(𝐴) the probability that the event 𝐴 will occur at each repetition of the random experiment. Remember that 𝐴 is said to have occurred if one element making up 𝐴 has occurred. In order to calculate or estimate the probability of an event 𝐴 there are two possibilities. In one approach an experiment can be performed a large number of times, and P(𝐴) can be approximated by the relative frequency with which 𝐴 occurs. In order to analyze random experiments we make the assumption that the conditions surrounding the trials remain the same, and are independent of one another. We hope 1
- 18. 2 SOME BACKGROUND ON PROBABILITY that some regularity or settling down of the outcome is apparent. The ratio the number of times a particular event 𝐴 occurs total number of trials is known as the relative frequency of the event, and the number to which it ap- pears to converge as the number of trials increases is known as the probability of an outcome within 𝐴. Where we have a finite sample space it might be reasonable to assume that the outcomes of an experiment are equally likely to occur as in the case, for example, in rolling a fair die or spinning an unbiased coin. In this case the probability of 𝐴 is given by P(𝐴) = number of elements of 𝑆 where 𝐴 occurs number of elements in 𝑆 . There are, of course, many ‘experiments’ which are not repeatable. Horse races are only run once, and the probability of a particular horse winning a particular race may not be calculated by relative frequency. However, a punter may form a view about the horse based on other factors which may be repeated over a series of races. The past form of the horse, the form of other horses in the race, the state of the course, the record of the jockey, etc., may all be taken into account in determining the probability of a win. This leads to a view of probability as a ‘degree of belief’ about uncertain outcomes. The odds placed by bookmakers on the horses in a race reflect how punters place their bets on the race. The odds are also set so that the bookmakers expect to make a profit. It is convenient to use set notation when deriving probabilities of events. This leads to 𝑆 being termed the universal set, the set of all outcomes: an event 𝐴 is a subset of 𝑆. This also helps with the construction of more complex events in terms of the unions and intersections of several events. The Venn diagrams shown in Figure 1.1 represent the main set operations of union (∪), intersection (∩), and complement (𝐴𝑐 ) which are required in probability. ∙ Union. The union of two sets 𝐴 and 𝐵 is the set of all elements which belong to 𝐴, or to 𝐵, or to both. It can be written formally as 𝐴 ∪ 𝐵 = {𝑥∣𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵 or both}. ∙ Intersection. The intersection of two sets 𝐴 and 𝐵 is the set 𝐴∩𝐵 which contains all elements common to both 𝐴 and 𝐵. It can be written as 𝐴 ∩ 𝐵 = {𝑥∣𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵}. ∙ Complement. The complement 𝐴𝑐 of a set 𝐴 is the set of all elements which belong to the universal set 𝑆 but do not belong to 𝐴. It can be written as 𝐴𝑐 = {𝑥 ∕∈ 𝐴}. So, for example, in an experiment in which we are interested in two events 𝐴 and 𝐵, then 𝐴𝑐 ∩ 𝐵 may be interpreted as ‘only 𝐵’, being the intersection of the complement of 𝐴 and 𝐵 (see Figure 1.1d): this is alternatively expressed in the difference notation 𝐵∖𝐴 meaning 𝐵 but not 𝐴. We denote by 𝜙 the empty set, that is the set which contains no elements. Note that 𝑆𝑐 = 𝜙. Two events 𝐴 and 𝐵 are
- 19. PROBABILITY 3 U U U A A A B B (a) (b) (c) U (d) B A Figure 1.1 (a) the union 𝐴 ∪ 𝐵 of 𝐴 and 𝐵; (b) the intersection 𝐴 ∩ 𝐵 of 𝐴 and 𝐵; (c) the complement 𝐴𝑐 of 𝐴: 𝑆 is the universal set; (d) 𝐴𝑐 ∩ 𝐵 or 𝐵∖𝐴 said to be mutually exclusive if 𝐴 and 𝐵 have no events in common 𝐴 and 𝐵 so that 𝐴 ∩ 𝐵 = 𝜙, the empty set: in set terminology 𝐴 and 𝐵 are said to be disjoint sets. The probability of any event satisfies however calculated the three axioms ∙ Axiom 1: 0 ≤ P(𝐴) ≤ 1 for every event 𝐴 ∙ Axiom 2: P(𝑆) = 1 ∙ Axiom 3: P(𝐴∪𝐵) = P(𝐴)+P(𝐵) if 𝐴 and 𝐵 are mutually exclusive (𝐴∩𝐵 = 𝜙) Axiom 3 may be extended to more than two mutually exclusive events, say 𝑘 of them represented by 𝐴1, 𝐴2, . . . , 𝐴𝑘 where 𝐴𝑖 ∩ 𝐴𝑗 = 𝜙 for all 𝑖 ∕= 𝑗. This is called a partition of 𝑆 if ∙ (a) 𝐴𝑖 ∩ 𝐴𝑗 = 𝜙 for all 𝑖 ∕= 𝑗, ∙ (b) 𝑘 ∪ 𝑖=1 𝐴𝑖 = 𝐴1 ∪ 𝐴2 ∪ . . . ∪ 𝐴𝑘 = 𝑆, ∙ (c) P(𝐴𝑖) > 0. In this definition, (a) states that the events are mutually exclusive, (b) that every event in 𝑆 occurs in one of the events 𝐴𝑖, and (c) implies that there is a nonzero probability that any 𝐴𝑖 occurs. It follows that 1 = P(𝑆) = P(𝐴1 ∪ 𝐴2 ∪ ⋅ ⋅ ⋅ ∪ 𝐴𝑘) = 𝑘 ∑ 𝑖=1 P(𝐴𝑖). Theorem ∙ (a) P(𝐴𝑐 ) = 1 − P(𝐴); ∙ (b) P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵) − P(𝐴 ∩ 𝐵). (a) Axiom 3 may be combined with Axiom 2 to give P(𝐴𝑐 ), the probability that the
- 20. 4 SOME BACKGROUND ON PROBABILITY complement 𝐴𝑐 occurs, by noting that 𝑆 = 𝐴 ∪ 𝐴𝑐 . This is a partition of 𝑆 into the mutually exclusive exhaustive events 𝐴 and 𝐴𝑐 . Thus 1 = P(𝑆) = P(𝐴 ∪ 𝐴𝑐 ) = P(𝐴) + P(𝐴𝑐 ), giving P(𝐴𝑐 ) = 1 − P(𝐴). (b) For any sets 𝐴 and 𝐵 𝐴 ∪ 𝐵 = 𝐴 ∪ (𝐵 ∩ 𝐴𝑐 ), and 𝐵 = (∩𝐵) ∪ (𝐵 ∩ 𝐴𝑐 ), in which 𝐴 and 𝐵 ∩ 𝐴𝑐 are disjoint sets, and 𝐴 ∩ 𝐵 and 𝐵 ∩ 𝐴𝑐 are disjoint sets. Therefore, by Axiom 3, P(𝑎 ∪ 𝐵) = P(𝐴) + P(𝐵 ∩ 𝐴𝑐 ), and P(𝐵) = P(𝐴 ∩ 𝐵) + P(𝐵 ∩ 𝐴𝑐 ). Elimination of P(𝐵 ∩ 𝐴𝑐 ) between these equations leads to P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵) − P(𝐴 ∩ 𝐵) (1.1) as required. Example 1.1. Two distinguishable fair dice 𝑎 and 𝑏 are rolled and the values on the uppermost faces noted. What are the elements of the sample space? What is the probability that the sum of the face values of the two dice is 7? What is the probability that at least one 5 appears? We distinguish first the outcome of each die so that there are 6 × 6 = 36 possible outcomes for the pair. The sample space has 36 elements of the form (𝑖, 𝑗) where 𝑖 and 𝑗 take all integer values 1, 2, 3, 4, 5, 6, and 𝑖 is the outcome of die 𝑎 and 𝑗 is the outcome of 𝑏. The full list is 𝑆 = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }, and they are all assumed to be equally likely since the dice are fair. If 𝐴1 is the event that the sum of the dice is 7, then from the list, 𝐴1 = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)} which occurs for 6 elements out of 36. Hence 𝑃(𝐴1) = 6 36 = 1 6 . The event that at least one 5 appears is the list 𝐴2 = {(1, 5), (2, 5), (3, 5), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 5)},
- 21. CONDITIONAL PROBABILITY AND INDEPENDENCE 5 which has 11 elements. Hence 𝑃(𝐴2) = 11 36 . Example 1.2 From a well-shuffled pack of 52 playing cards a single card is randomly drawn. Find the probability that it is a heart or an ace. Let 𝐴 be the event that the card is an ace, and 𝐵 the event that it is a heart. The event 𝐴∩𝐵 is the ace of hearts. We require the probability that it is an ace or a heart, which is P(𝐴 ∪ 𝐵). However, since one of the aces is a heart the events are not mutually exclusive. Hence, we must use eqn (1.1). It follows that the probability that an ace is drawn is P(𝐴) = 4/52, the probability that a heart is drawn is P(𝐵) = 13/52 = 1/4, the probability that the ace of hearts is drawn is P(𝐴 ∩ 𝐵) = 1/52. From (1.1) P(𝐴 ∪ 𝐵) = P(𝐴) + P(𝐵) − P(𝐴 ∩ 𝐵) = 4 52 + 1 4 − 1 52 = 16 52 = 4 13 . This example illustrates events which are not mutually exclusive. The result could also be obtained directly by noting that 16 of the 52 cards are either hearts or aces. In passing note that 𝐴 ∩ 𝐵𝑐 is the set of aces excluding the ace of hearts, whilst 𝐴𝑐 ∩ 𝐵 is the heart suit excluding the ace of hearts. Hence P(𝐴 ∩ 𝐵𝑐 ) = 3 52 , P(𝐴𝑐 ∩ 𝐵) = 12 52 = 3 13 . 1.3 Conditional probability and independence If the occurrence of an event 𝐵 is affected by the occurrence of another event 𝐴 then we say that 𝐴 and 𝐵 are dependent events. We might be interested in a random ex- periment with which 𝐴 and 𝐵 are associated. When the experiment is performed, it is known that event 𝐴 has occurred. Does this affect the probability of 𝐵? This prob- ability of 𝐵 now becomes the conditional probability of 𝐵 given 𝐴, which is now written as P(𝐵∣𝐴). Usually this will be distinct from the probability P(𝐵). Strictly speaking, this probability is conditional since we must assume that 𝐵 is conditional on the sample space occurring, but it is implicit in P(𝐵). On the other hand the con- ditional probability of 𝐵 is restricted to that part of the sample space where 𝐴 has occurred. This conditional probability is defined as P(𝐵∣𝐴) = P(𝐴 ∩ 𝐵) P(𝐴) , P(𝐴) > 0. (1.2) In terms of counting, suppose that an experiment is repeated N times, of which 𝐴 occurs N(𝐴) times, and 𝐴 given by 𝐵 occurs N(𝐵 ∩ 𝐴) times. The proportion of times that 𝐵 occurs is N(𝐵 ∩ 𝐴) N(𝐴) = N(𝐴 ∩ 𝐵) N N N(𝐵) ,
- 22. 6 SOME BACKGROUND ON PROBABILITY which justifies (1.2). If the probability of B is unaffected by the prior occurrence of 𝐴, then we say that 𝐴 and 𝐵 are independent or that P(𝐵∣𝐴) = P(𝐵), which from above implies that P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵). Conversely, if P(𝐵∣𝐴) = P(𝐵), then 𝐴 and 𝐵 are independent events. Again this result can be extended to 3 or more independent events. Example 1.3 Let 𝐴 and 𝐵 be independent events with P(𝐴) = 1 4 and P(𝐵) = 2 3 . Calculate the following probabilities: (a) P(𝐴 ∩ 𝐵); (b) P(𝐴 ∩ 𝐵𝑐 ); (c) P(𝐴𝑐 ∩ 𝐵𝑐 ); (d) P(𝐴𝑐 ∩ 𝐵); (e) P((𝐴 ∪ 𝐵)𝑐 ). Since the events are independent, then P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵). Hence (a) P(𝐴 ∩ 𝐵) = 1 4 ⋅ 2 3 = 1 6 . (b) The independence 𝐴 and 𝐵𝑐 follows by eliminating P(𝐴 ∩ 𝐵) between the equations P(𝐴 ∩ 𝐵) = P(𝐴)P(𝐵) = P(𝐴)[1 − P(𝐵𝑐 )] and P(𝐴) = P[(𝐴 ∩ 𝐵𝑐 ) ∪ (𝐴 ∩ 𝐵)] = P(𝐴 ∩ 𝐵𝑐 ) + P(𝐴 ∩ 𝐵). Hence P(𝐴 ∩ 𝐵𝑐 ) = P(𝐴)P(𝐵𝑐 ) = P(𝐴)[1 − P(𝐵)] = 1 4 (1 − 2 3 ) = 1 12 . (c) Since 𝐴𝑐 and 𝐵𝑐 are independent events, P(𝐴𝑐 ∩ 𝐵𝑐 ) = P(𝐴𝑐 )P(𝐵𝑐 ) = [1 − P(𝐴)][1 − P(𝐵)] = 3 4 ⋅ 1 3 = 1 4 . (d) Since 𝐴𝑐 and 𝐵 are independent events, P(𝐴𝑐 ∩ 𝐵) = P(𝐴𝑐 )P(𝐵) = [1 − 1 4 ]2 3 = 1 2 . (e) P((𝐴 ∪ 𝐵)𝑐 ) = 1 − P(𝐴 ∪ 𝐵) = 1 − P(𝐴) − P(𝐵) + P(𝐴 ∩ 𝐵) by (1.1). Hence P((𝐴 ∪ 𝐵)𝑐 ) = 1 − P(𝐴) − P(𝐵) + P(𝐴)P(𝐵) = 1 − 1 4 − 2 3 + 1 6 = 1 4 . Example 1.4. For three events 𝐴, 𝐵, and 𝐶, show that P(𝐴 ∩ 𝐵∣𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵∣𝐶), where P(𝐶) > 0. By using (1.2) and viewing 𝐴 ∩ 𝐵 ∩ 𝐶 as (𝐴 ∩ 𝐵) ∩ 𝐶 or 𝐴 ∩ (𝐵 ∩ 𝐶), P(𝐴 ∩ 𝐵 ∩ 𝐶) = P(𝐴 ∩ 𝐵∣𝐶)P(𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵 ∩ 𝐶). Hence P(𝐴 ∩ 𝐵∣𝐶) = P(𝐴∣𝐵 ∩ 𝐶) P(𝐵 ∩ 𝐶) P(𝐶) = P(𝐴∣𝐵 ∩ 𝐶)P(𝐵∣𝐶) by (1.2) again. A result known as the law of total probability or the partition theorem will
- 23. DISCRETE RANDOM VARIABLES 7 A1 A2 A5 A4 A3 S Figure 1.2 Schematic set view of a partition of 𝑆 into 5 events 𝐴1, . . . , 𝐴5. be used extensively later, for example, in the discrete gambler’s ruin problem (Sec- tion 2.1) and the Poisson process (Section 5.2). Suppose that 𝐴1, 𝐴2, . . . , 𝐴𝑘 repre- sents a partition of 𝑆 into 𝑘 mutually exclusive events in which, interpreted as sets, the sets fill the space 𝑆 but with none of the sets overlapping. Figure 1.2 shows such a scheme. When a random experiment takes place one and only one of the events can take place. Suppose that 𝐵 is another event associated with the same random experiment (Fig- ure 1.2). Then 𝐵 must be made up of the sum of the intersections of 𝐵 with each of the events in the partition. Some of these will be empty but this does not matter. We can say that 𝐵 is the union of the intersections of 𝐵 with each 𝐴𝑖. Thus 𝐵 = 𝑘 ∪ 𝑖=1 𝐵 ∩ 𝐴𝑖, but the significant point is that any pair of these events is mutually exclusive. It follows that P(𝐵) = 𝑘 ∑ 𝑖=1 P(𝐵 ∩ 𝐴𝑖). (1.3) Since, from equation (1.2), P(𝐵 ∩ 𝐴𝑖) = P(𝐵∣𝐴𝑖)P(𝐴𝑖), equation (1.3) can be expressed as P(𝐵) = 𝑘 ∑ 𝑖=1 P(𝐵∣𝐴𝑖)P(𝐴𝑖), which is the law of total probability or the partition theorem. 1.4 Discrete random variables In most of the applications considered in this text, the outcome of the experiment will be numerical. A random variable usually denoted by the capital letters 𝑋, 𝑌 , or 𝑍, say, is a numerical value associated with the outcome of a random experiment.
- 24. 8 SOME BACKGROUND ON PROBABILITY If 𝑠 is an element of the original sample space 𝑆, which may be numerical or sym- bolic, then 𝑋(𝑠) is a real number associated with 𝑠. The same experiment, of course, may generate several random variables. Each of these random variables will, in turn, have sample spaces whose elements are usually denoted by lower case letters such as 𝑥1, 𝑥2, 𝑥3, . . . for the random variable 𝑋. We are now interested in assigning proba- bilities to events such as P(𝑋 = 𝑥1), the probability that the random variable 𝑋 is 𝑥1 and P(𝑋 ≤ 𝑥2), the probability that the random variable is less than or equal to 𝑥2. If the sample space is finite or countably infinite on the integers (that is, the ele- ments 𝑥0, 𝑥1, 𝑥2, . . . can be counted against integers, say 0, 1, 2, . . .) then we say that the random variable is discrete. Technically, the set {𝑥𝑖} will be a countable subset 𝒱, say, of the real numbers ℛ. We can represent the {𝑥𝑖} generically by the variable 𝑥 with 𝑥 ∈ 𝒱. For example, 𝒱 could be the set {0, 1 2 , 1, 3 2 , 2, 5 2 , 3, . . .}. In many cases 𝒱 consists simply of the integers or a subset of the integers, such as 𝒱 = {0, 1} or 𝒱 = {0, 1, 2, 3, . . .}. In the random walks of Chapter 3, however, 𝒱 may contain all the positive and neg- ative integers . . . − 3, −2, −1, 0, 1, 2, 3, . . .. In these integer cases we can put 𝑥𝑖 = 𝑖. The function 𝑝(𝑥𝑖) = P(𝑋 = 𝑥𝑖) is known as the probability mass function. The pairs {𝑥𝑖, 𝑝(𝑥𝑖)} for all 𝑖 in the sample space define the probability distribution of the random variable 𝑋. If 𝑥𝑖 = 𝑖, which occurs frequently in applications, then 𝑝(𝑥𝑖) = 𝑝(𝑖) is replaced by 𝑝𝑖. Since the 𝑥 values are mutually exclusive and exhaustive then it follows that ∙ (a) 0 ≤ 𝑝(𝑥𝑖) ≤ 1 for all 𝑖, ∙ (b) ∞ ∑ 𝑖=0 𝑝(𝑥𝑖) = 1, or in generic form ∑ 𝑥∈𝒱 𝑝(𝑥) = 1, ∙ (c) P(𝑋 ≤ 𝑥𝑘) = 𝑘 ∑ 𝑖=0 𝑝(𝑥𝑖), which is known as the distribution function. Example 1.5. A fair die is rolled until the first 6 appears face up. Find the probability that the first 6 appears at the 𝑛-th throw. Let the random variable 𝑁 be the number of throws until the first 6 appears face up. This is an example of a discrete random variable 𝑁 with an infinite number of possible outcomes {1, 2, 3, . . .} . The probability of a 6 appearing for any throw is 1 6 and of any other number appearing is 5 6 .
- 25. CONTINUOUS RANDOM VARIABLES 9 Hence the probability of 𝑛 − 1 numbers other than 6 appearing followed by a 6 is P(𝑁 = 𝑛) = (5 6 )𝑛−1 (1 6 ) = 5𝑛−1 6𝑛 , which is the probability mass function for this random variable. 1.5 Continuous random variables In many applications the discrete random variable, which for example might take the integer values 1, 2, . . ., is inappropriate for problems where the random variable can take any real value in an interval. For example, the random variable 𝑇 could be the time measured from time 𝑡 = 0 until a light bulb fails. This could be any value 𝑡 ≥ 0. In this case 𝑇 is called a continuous random variable. Generally, if 𝑋 is a continuous random variable there are mathematical difficulties in defining the event 𝑋 = 𝑥: the probability is usually defined to be zero. Probabilities for continuous random variables may only be defined over intervals of values as, for example, in P(𝑥1 < 𝑋 < 𝑥2). We define a probability density function (pdf) 𝑓(𝑥) over −∞ < 𝑥 < ∞ which has the properties: ∙ (a) 𝑓(𝑥) ≥ 0, (−∞ < 𝑥 < ∞); ∙ (b) P(𝑥1 ≤ 𝑋 ≤ 𝑥2) = ∫ 𝑥2 𝑥1 𝑓(𝑥)𝑑𝑥 for any 𝑥1, 𝑥2 such that −∞ < 𝑥1 < 𝑥2 < ∞; ∙ (c) ∫ ∞ −∞ 𝑓(𝑥)𝑑𝑥 = 1. A possible graph of a density function 𝑓(𝑥) is shown in Figure 1.3. By (a) above the curve must remain nonnegative, by (b) the probability that 𝑋 lies between 𝑥1 f(x) x x1 x2 Figure 1.3 A probability density function. and 𝑥2 is the shaded area, and by (c) the total area under the curve must be 1 since P(−∞ < 𝑋 < ∞) = 1. We define the (cumulative) distribution function (cdf) 𝐹(𝑥) as the probability
- 26. Another Random Scribd Document with Unrelated Content
- 27. l6o MOST-FOR-YOUR-MONEY COOK BOOK Bits of Cooked Vegetables. Mix into omelet or egg scramble; make baked vegetable hash of them, sprinkled with crumbs and grated cheese. Fish Heads and Bones. Set to simmer in cold water and use the broth, after straining through cloth, for fish sauces, or soup, or boil a whole fish in it. Cold Boiled Fish. Shred and mix with potato or other vegetable salad. Cold Fried Fish. Soak whole pieces for 2-3 hours in salad dressing. Serve with onion slices and a green garnish. Leftover Spaghetti and Other Pastes. Cut up, reheat, and use for stretching egg dishes. Soft bread crumbs and leftover potatoes, either mashed or sliced, serve the same purpose. Leftover Gravy. Add to the next gravy, or heat leftover vegetables in it. Bones and Meat Trimmings. Into the soup pot, of course. Cold Fried Bacon. Reheat, crumble into a vegetable dish, or into the soup just before serving; or mix with scrambled eggs. Poultry Feet and Trimmings. Clean, scald and trim feet, simmer with trimmings, and use for gravy or soup; reduced, this broth makes the most delicious aspic. THE BEST ASPIC YOU EVER TASTED! And the Most Economical Use fresh chicken giblets, also the wing tips, neck and feet. Prepare feet by immersing 1-2 minutes in boiling water and removing skin and nails. Put feet with neck, wing tips, liver, heart and gizzard into a pan, add a little parsley, carrot, onion and 2 cups of water for the parts of each chicken, simmer gently for 2>^ hours, which will reduce the broth to one-half. Flavor with lemon or strong sweet wine such as sherry, or both. It doesn't pay to use less than the parts of 3-4 chickens, but since nearly everybody throws away the feet and giblets, these can be bought separately from butchers who serve hotels. It shouldn't be either difficult or expensive to get the makings of this exceptional aspic which is even cheaper than making it of gelatine or calves* feet, which aren't stocked commonly, except by
- 28. HANDY HINTS l6l kosher butchers. And there's no comparison between this tastyaspic and one made of gelatine, or calves* feet. Fowl Fat. Chicken or turkey fat, reduced to oil, takes the place of melted butter and makes a tasty salad oil as well. Also makes a swell sandwich spread. Any Part Except the Quack. The meat packers got rich by utilizing every part of the pig down to the squeal, and we can keep from getting poorer faster by economizing on the duck and goose in the same way, using every part of the duck except the quack, and of the goose down to its hiss: 1st: save the tongue, for that's the gourmet's pick of either duck or goose, throw away the bill and the eyes, but use all the rest of the head, neck, tip ends of wings, all giblets and trimmings of skin and fat. The feet must be blanched and skinned and the toe nails discarded. Wash everything well and put in a pan with a quart of salted water, carrot, the roots of celery and of parsley. Cook until the toughest piece (probably the gizzard) begins to get tender, then toss in a cup of rice, some celery and two beef cubes dissolved in a cup of hot water. By the time the rice is cooked most of the liquid will be absorbed and you'll have a fine dish at the cost of what most people throw away. Pope's Noses. The Pope's nose, which is "the part that goes over the fence last," is the tastiest tidbit of any fowl, especially chicken and turkey, so don't by any chance throw that away. VINEGAR A clove of garlic in a bottle of almost any kind of vinegar peps it up most pleasingly, that is, if you like garlic, and if your vinegar is made honestly — not out of wood shavings. It's good to have a stock of different-flavored vinegars on hand to vary the savor of mayonnaise and other sauces. Get good country cider vinegar if you can, and buy it by the gallon instead of in those tricky small chain store bottles made like magnifying bar glasses to make a very little look like a lot. Fresh or dried herbs are cheap in every "foreign" market and their full flavor can be added to a pint or quart of your bulk vinegar by using such combinations as the following:
- 29. 1 62 MOST-FOR-YOUR-MONEY COOK BOOK In one pint of vinegar put i tablespoon each of minced water cress or highland cress, shallot, chervil and rosemary, with one clove of garlic, and let it steep in the sun until it's full flavored. In yi pint of red wine vinegar steep 4 tablespoons shredded cucumber rind, i tablespoon grated horseradish, i tablespoon minced chives or shallots. Or, I dessertspoon each of various seeds, such as celery, dill and caraway, made sprightly with minced cress, parsley and 2 capsicums. You can make your own wine vinegar by letting cheap California wine sour, adding the "mother'* from any old vinegar or a little yeast to start the acetic acid. It's always most satisfactory to make your own vinegar. Then you'll know what went into it. Apple juice, cider or peaches are best for this, but the following vegetable vinegars are cheap and satisfactory: VEGETABLE VINEGARS Tomato. Press juice from ripe tomatoes into a large vessel or crock, leave uncovered in a moderately warm place and it will quickly turn to vinegar. Add Yi cup molasses to every quart of juice if you want an extra-acid vinegar. In season, tomatoes should be almost as cheap as the dirt they grow in, especially in city markets on Saturday night, when all ripe produce has to be sold out because it won't keep until Monday. Beet. Run a bushel of washed beets through a grinder 2 or 3 times to extract all the juice — put juice in open cask or crock covered against insects and light, set in the sun for 2-3 weeks and by that time you'll have several gallons of good tasty vinegar. Potato. To every gallon of water that potatoes have been boiled in put half a cake of yeast and a pound of brown sugar. This will turn to vinegar in less than a month, but it's not as nicely flavored as most. It serves, however, for cooking and making fresh cucumber pickles without salt, simply by putting in cucumbers cut fresh from the vines and adding dill if you wish. And the chances are that this potato peel product will be as good or better than the average store article. Nasturtium. Bruise Nasturtium flowers, cover with cold vinegar,
- 30. HANDY HINTS 1 63 add I shallot, and yi garlic clove to each quart. Set aside for 1 months. Strain, add yi ounce cayenne and >2 ounce salt. MINT This fresh herb is so refreshing, especially in lamb sauce and lemonade, that a nickel bunch of it should always be on hand in the kitchen to use as the most-for-your-money flavoring in stews, sauces and beverages. It is packed so full of flavor that a sprig or two go farther than a whole handful of watercress or other lively relish. To our mind it's the most economical flavoring we know, and besides, mint seems exactly suited to our All-American taste. GARDEN GARNISHES AND MISCELLANIES Celery Curls. Cut stalks in 2-inch lengths and with a sharp knife cut ^-^ slits about i-inch deep in end of each piece and throw into ice water several hours before using, so they'll be nicely curled. Radishes. Remove tips and retain stem with sufficient green leaves to be attractive and make a good handle for eating them. Keep in ice water until wanted. Serve on chipped ice. Parsley Root. Many people throw away the roots of parsley after using the tops for garnishing, but some varieties have a fine tuber that's as savory as celery root for seasoning soups and stews; so that should never be thrown away. Capers. Since everybody relishes a relish, always have a bottle of capers on hand, for snacks and seasoning. A bottle needn't cost more than 20^ and can be used to pep up a plate of cold cuts, sandwiches, all sorts of tidbits, at a cost of less than a cent a person, for a bottle should be enough to spice half a dozen goodsized platters. In some places you can buy capers cheaper in bulk, or if you can get hold of nasturtium pods they're just as good. You pickle them simply by dropping them into a bottle of vinegar. The green seed-pods of radishes also take the place of capers, at no cost at all. Peas. A dish of French green peas is almost a meal in itself and is always served as a separate course over there. The secret is in cooking freshly shelled peas an hour in a little butter, with some bits of bacon, lettuce leaves, spring onions and parsley. The flavors
- 31. 164 MOST-FOR-YOUR-MONEY COOK BOOK of lettuce and peas not only complement each other, but the lettuce serves to put on top of the peas to keep them savorily smothered while cooking. Likewise, to perk up canned peas, cover them with lettuce leaves, add a little green onions and butter, or bacon, or both. Pepper Skins. Since these are too tough to eat, before using peppers as a garnish or in salad you can easily remove the skins by holding them over flame on a fork until skin begins to blister; then peel it off. Water. Water is Flavor Enemy No. i. Too much washing of meat and vegetables removes a great deal of their flavor. There's an old English song that advises: "But don't let the water get into the wine." The French, after washing lettuce, swing it in a basket until it's bone dry and able to absorb the oil of the dressing, while anybody who knows his mushrooms never washes them, but wipes them clean with a moist cloth. Cucumbers. Lots of people have an allergy to fresh cucumbers and few of us can get along without soaking them for a while in salted water. Although they're usually sliced like a loaf of bread, the way to get their full flavor is to slit lengthwise in quarters and instead of using black pepper, sprinkle with paprika. Fresh cool cucumber makes a fine contrast to eat out of hand with a dish of hot vegetables. Foods Free for the Picking. Even city dwellers have access to suburban fields and forests where there are herbs, nuts, mushrooms, wild berries, fruits and even edible flowers free for the picking. It pays to learn about the good mushrooms, for instance, and gather a sack or two of these neglected foods on a holiday. You can have a feed or two of the fresh ones and then dry the rest by stringing them like necklaces and hanging in a dry place. Because of their light weight when dried these cost a dollar and more a pound in the markets that stock them, and there's nothing so tasty in stews and soups. So if you don't want to trouble to gather and dry them yourself, you can buy a dime's worth, soak them overnight and give that old mushroom flavor to a whole gallon of soup. Pork and Mutton. As with oysters, the "R" months are best for eating both pork and mutton which are really in their prime only in October and November.
- 32. HANDY HINTS 1 65 Ripe Olives. Oily black olives are a gift to any cook who'll learn to use them. They cost very little in foreign neighborhood markets and are bursting with rich flavor which imparts piquance to many a stew and salad. Spaghetti. On English menus Italian paste products, spaghetti, macaroni and vermicelli are listed under "vegetables"; and while we do not think of them in this way, it's good to remember, especially in winter, that if you haven't any vegetables at hand to give a lift to a meat stew, sometimes lightly cooked Italian pastes, especially if made of good hard wheat, take the place of vegetables. Juniper Berries. These peppy dried berries which we think of mostly for flavoring gin do just as cheerful a job when they're crushed and sprinkled over stews and hash. Popcorn. Buy the seed by the pound, pop it yourself in one of those mosquito-wire poppers and season with salt and melted butter, or do it cheaper by popping in lard in an iron kettle and then salting. It's the finest garnish we know for cream soups, where a few crunchy, buttery, snowy kernels scattered on at the finish make all the diflference in the world. It's great to eat by the handful with a glass of milk or to use as a cereal at a fraction of the cost of puffed grains put up by the breakfast food racketeers, or packageers. Try it in a bowl of milk in place of bread, and savor everything by crumbling in bits of fine old store cheese. Makes a great snack at anytime and is just suited to the palate of this cornfed country. Peanuts. Cheap and handy for making anything from peanut soup to candy. In fact. Dr. George W. Carver of Tuskegee Institute not only extracted sweet milk and rich cream equal to the cow's from this lowly nut that's chockful of food value, but he gave a five-course luncheon to food experts, consisting of 14 dishes in every one of which the chief ingredient was peanuts. And then he compiled 105 peanut recipes, ranging from peanut bread, rolls, cookies, wafers, muffins, doughnuts, to cake, salad and candy bars. Peanut Butter. Many commercial peanut butters are ruined in flavor by draining off^ the peanut oil to sell for a good price and then substituting cottonseed oil. If you don't know where to get honest peanut butter, which is the handiest sort of thing for help
- 33. l66 MOST-FOR-YOUR-MONEY COOK BOOK ing the flavor of all kinds of dishes, write to Llano Cooperative Colony, Newllano, Louisiana, and they'll supply the finest we ever tasted, at a price much lower than any chain store offers. Piping Hot. The recent success of sizzling platters, whose virtue is that the meat is still hissing on a metal platter when brought to table, proves that we like our hot dishes piping. This can be done with stews and vegetables as well, bringing them on bubbling in thick earthenware dishes, so they're as attractive as a savory Lobster Newburg or Welsh Rabbit made right on the table. It's also good to remember to heat the diner's plate in cold weather and in summer to chill plates before serving salad, cold consommes and such things on them, just as a good bartender chills his beer steins before using. HOMEMADE FIRELESS COOKERS We seldom hear of fireless cookers these days, but at one time not so long ago, they were a part of regulation kitchen equipment, and they cut dollars off the yearly fuel bills. World War propaganda further popularized them, for then all housewives were urged to save coal, not so much for their own account as for the dear Allies. Wheatless and meatless days, and fireless cookers for pure patriotism, with never a thought on the part of either war profiteers or their helpless fellow citizens that after-the- war-depression would bring enforced wheatless weeks and meatless months to millions. Metals, which are wasted in peace times on all sorts of useless contraptions, had to be conserved to their death dealing ends. So the press carried instructions for making fireless cookers at home. All one needed was a wooden box or paper carton, and a lot of old newspapers to insulate it, layers of paper fitted into the bottom of the box and around the sides, with a cylindrical hole left in the center to receive a boiling pot of soup or stew; then wads of paper on top to hold in all the heat for hours. An excellent device for long, slow cooking of cheap foods. Dried beans, peas, and lentils, tendered in their unbroken skins; and cereals, started the night before, are still hot at breakfast time and have attained a jellylike and delicate consistency which only many hours of low heat can give.
- 34. HANDY HINTS 1 67 These economical, practical cookers will save coal now, just as they did then, and are well worth reviving. And before we leave the wheatless, meatless and eatless days when owner patriots urged wage-slaves' wives to half starve their families and "Win the War in the Kitchen," we'd like to record the fact that one of these Official Recipe Books was published under the chairmanship of Samuel Insull. SEASONING SECRETS A row of jars and bottles of home prepared seasonings is a neverending inspiration on those days when one is distracted with thinking up new ways to vary the menus. Herbs such as rosemary, savory, dill, marjoram and tarragon can be collected, one at a time, as they appear on the pushcarts. Dried and kept away from the air, they will last a year. The leaves from a bunch of celery will dry also, keeping their natural color and full flavor if laid on a pie tin in an open warm oven, or hung on a string over radiator or behind stove, until brittle. Piled up loosely in a covered glass jar, these will always provide fresh celery seasoning for the soup and stew pot, or roasting pan, for sauces and for gravies. Parsley cannot be successfully dried for commerce, but for one's own kitchen it will dry as well as celery and take the place of green parsley, especially if it is refreshed in cold water 15 minutes before using. Basil, too, can be dried at home in the same way. But living plants of basil or sweet marjoram, procured from florist or vegetable vendor, will grow in pots on any window sill, bravely putting out more leaves to take the place of the ones continually plucked oflF. In summer, fire-escapes in foreign sections of our cities always display tin cans and boxes of these two sturdy plants which are taken inside in winter. With this store of herbs should go a string of little hot red peppers, likewise dried out in the open oven, so they will not spoil. An unopened pod is not too much for a pot of soup. If only a portion is used, be sure the seeds are discarded. And be sure to take it out before serving, or some person will get more than his share of hotness.
- 35. l68 MOST-FOR-YOUR-MONEY COOK BOOK There is nothing like herb variety for relieving food monotony and giving a family the feeling that they are eating something different all the time. Most herbs are both appetizers and aids to digestion besides, and thus they may be used freely without fear of injury to health. Although most of them are so strong that a little goes a long way, there are a few, like the faithful basil that grows in a pot, whose leaves can be scattered through green salads or, minced, may be put into the salad dressing until it is actually green. Basil is the best herb of all with tomatoes, either raw or cooked, and Italians like to lay a few leaves of it on a platter of spaghetti with tomato sauce, for each person to tear up a leaf and mix it in, or nibble at it whole, while eating. The time to lay in orange and lemon flavorings is when these are cheapest. Before cutting the fruit, carefully pare off the outside yellow covering of the skin in a long unbroken spiral with as little of the white sticking to it as possible. Toss the spirals over a string stretched in a warm place. When the moisture has dried out, but before the skin has become crisp, pack away in a glass jar. Eventually there will be two jars full, one of lemon and the other of orange peel, ready for the season when fresh citrus fruit is scarce and expensive. Two inches or so broken off will flavor soup, or pudding or pie, will give a zest to applesauce or other cooked fruits and is handy in concocting drinks. For use in certain cakes, cookies and puddings it is better to candy these peels. CANDIED ORANGE AND LEMON PEELS After juice has been reamed from oranges or lemons, tear out the membranes. Soak over night in water salted with i tablespoon of salt to the quart. Drop into fresh water and drain. With a spoon scrape off white inside, and cut with scissors into narrow strips. Mix yi cup of water with i cup sugar, add the peel and cook until the strips look clear. Drain, lay on a plate to cool, then roll in granulated sugar, letting as much sugar as possible stick to them. When dry pack away in jars or tin boxes lined with paper. If they become too dry after a time, warm them in the oven.
- 36. HANDY HINTS 1 69 CANDIED GRAPEFRUIT PEEL Grapefruit peel is so bitter that it should be parboiled for 15 minutes after being taken from the salt water; and some people parboil, drain, parboil a second time and drain, before cooking with sugar. After parboiling follow the recipe for Candied Orange and Lemon Peels. There's a whole book about using citrus peels, if you're interested. Mrs. Florence Gilson Barton wrote and published it in San Bernardino, California, in 1928. It's called ne California Orange Cook Book, Complete and Explicit Directions for the Making of Candied and Glace Fruits, Jellies, Marmalades; Orange and Grapefruit Rinds in Decorative Forms and Preserves. And while you're at it, experiment with the rinds of Persian limes and kumquats.
- 37. XXIII. The Pick of the Pushcart We've made our very best vegetable and fruit buys, over a period of thirty years, off pushcarts in the poorest sections of cities all over the world, not counting the unbeatable bargains on the tropical fruit wharves of Bahia and the raspberries and melons sold in season on every Moscow street corner and in the unbelievably bounteous Gastronomes throughout the Soviet. The reason for pushcart plenty is obvious, especially in a huge city like New York. In this broad land some fruit is always in season, and the pick of all crops comes to this highest priced market; so, since the best of it ripens en route, every day something or other arrives in the commission merchant's hands that's prime but risky for big dealers to hold for a gamble. He's either got to sell it out at once for what he can get, or throw it away. Shipments with slight imperfections also can't be resold to Park Avenue Bon Voyage basket fillers, so they're picked up by the haggard, hoarse street- hawkers who haunt Washington Market at dawn, with only ten or twenty dollars to invest in quick merchandise to load up pushcarts or a bag-of-bones horse and wagon they rent for from 50^ to ^3.00 a day. Without capital they can't hold out for prices as the chain stores do, so they yell and sell in Paddy's market, east side marts and on the streets, and if you know your artichokes, fresh figs, pomegranates or honeydews, the very best fruit value you can get is — the pick of the pushcart. The one sure quality test in picking out any kind of fruit is its 170
- 38. THE PICK OF THE PUSHCART I7I weight in comparison to its size. The heavier, the more mature, hence fullest of juice and flavor. Grapefruit offers an easy beginning; heft one in each hand and soon you'll come across a specimen as big as your head yet light as a balloon (please don't transpose this), while in the other hand youVe got one just half the size but heavy as a cannon ball. The heavy one is naturally the best buy, for the lighter one just has to be thick skinned and puffy — no juice ! In picking out any fruit remember it's already been handled enough by picker and packer, so don't pinch it to test its ripeness. Let the dealer do that, and he will, if his stuff is worth a darn. Always put it up to the seller to prove that his fruit is worth buying. If he won't do that, buy from one who will. Pinching, of course, bruises any fruit and starts decay, so you can't blame a dealer for getting sore when an amateur ruins half a dozen avocados or peaches by amateur poking which makes soft spots that quickly lead to waste. With berries, melons and smaller fruits, if the seller has something worth buying he's glad to give you a sample taste. If not, beware. In buying berries always ask the seller to turn them out in his cupped hands or pour them into another box to show you that the bottom ones are fair-sized, sound and dry, not mildewed, unpalatable runts. Naturally, you'll have to allow a little for those packed far down out of sight, since cheating has become an accepted practice in our competitive society. Even the consumer has been taught to expect fruits and vegetables to get smaller and gnarlier as he digs down beneath the top "show" layers. Whether you're from Missouri or not, **show me" is the buyer's slogan; so look at the bottom always to make sure it isn't a false one built up like the bottom of a post-prohibition beer stein, or that the fruits beneath are not entirely worthless, and when buying by weight make sure there's no concealed brick or horseshoe. Probably today spare-tires are used to make up the weight, but when a father of ours used to buy butter wholesale his testing plugger often struck fire on a "thrown" plough-horse shoe. And for God's sake don't be 100% American and embarrassed about looking for flaws; you can
- 39. be sure the dealer has done it thoroughly in the first place and has refused to pay full price for fakes. So take a
- 40. lyi MOST-FOR-YOUR-MONEY COOK BOOK tip from him and if the stuff is of lower value than he asks, don*t pay the asking price like a meek little lamb that loves to be shorn. Just tell him plainly what it's worth to you. Never be afraid to bargain. We are, in fact, the only people in the world who take it on the chin by paying fixed prices for bad or good without a peep. We're dumb as mud-puppies about protecting ourselves in the most vital concerns of life. Advertising, movies and such models of exploitation have made us think that maybe it's more elegant to just sit down at the phone and ask the butcher "Otto, how's your liver today ? Okay, send me two pounds." We're ashamed to be seen out with a market basket and pretty soon we'll take to letting our little fingernails grow long like Chinese Rockefellers, to prove that we don't live by using our own hands. Recently organized labor has grown weary of the owner's old "take it or leave it attitude" and begun to bargain with the boss, so labor's helpmeet can help meet the high-cost of living by bargaining determinedly with all food-choker-off-ers and if they don't give in, she can organize and pull a boycott, a sort of sitdown buyers' strike. (And while the following is a bit off the subject we can't resist putting in about that Soviet foreman in charge of shipping lumber. In competitive days under the Czar he'd learned to load ships light by building at night a false bottom in the ship to be loaded. Although under Socialism he had nothing to gain by this, the habit was so fixed in him that he continued cheating comradely consumers' co- ops and couldn't be made to see the light until he'd been arrested three or four times. Likewise, the leading purveyor of fruit juice drinks under the Czar held on to his recipes for palatable adulterations until the Soviet showed him up by going back to the root of the matter and making drinks of pure juice alone. And finally these fine drinks aroused the respect of the old drink- waterer; so he went to work for the beverage trust, adding his skill and technique to making what are now the finest, purest fruit drinks in the world.) Though our own Government supports the competitive system which carries its share of germs of capitalist decay which one day will
- 41. destroy the whole barrel, it gives honest advice about sharp practice and tells the consumer how to get his money's worth if he's
- 42. THE PICK OF THE PUSHCART 1 73 smart. So send a nickel (in coin, not stamps, and don*t ask us why the Government refuses the sticky little lozenges of paper it prints and sells for postal service at a price much higher here than in any country we know of) to the U. S. Department of Agriculture in Washington for Miscellaneous Publication No. 167, ^ Fruit and Vegetable Buying Guide for Consumers by R. G. Hill. We have used this bulletin with profit in our own summary of how to get your money's worth in buying fruits, but there's much more detailed information than we can pack in half a dozen pages: Apples. An apple a day may not keep the doctor away, but it'll keep almost anybody from falling for the phoney laxative ads. Since there are 500 different kinds on the market, don't buy just "apples." The kind we get in the East called Macintosh are a little more expensive, but we use their excellent flavor and juicy snap to judge all others. No apple should be too ripe, bruised or spotted. Medium-sized, thin-skinned ones are apt to be juiciest. And now that workers are demanding something more than the core, it's a good idea to list the most popular kinds and the seasons when they're cheapest. The Consumers Union's Buying Guide has summed up the apple situation for us, as follows: Favorite cooking apples are those having a slightly tart taste (Wealthy, Jonathan, Willow Twig, and Rome Beauty). Grimes Golden, Delicious, and Stayman Winesap are excellent dessert apples — but let us add that the Macintosh is best of all. September — Gravestein, Wealthy October — Jonathan, Grimes Golden, Delicious November — Jonathan, Spitzenberg, Delicious, King, Twenty Ounce December — Jonathan, Spitzenberg, Stayman Winesap, Delicious, Northern Spy, Rome Beauty January — Spitzenberg, Stayman Winesap, Northern Spy, Rome Beauty February — Baldwin, Rhode Island Greening, Yellow Newton, Stayman Winesap March — Yellow Newton, Stayman Winesap, Willow Twig April and May — Stayman Winesap, Willow Twig, Yellow Newton.
- 43. 174 MOST-FOR-YOUR-MONEY COOK BOOK So don't be satisfied with just "apples," like those recipe writers who say "add a cup offish" without saying what kind and knowing why that kind is best. No summer apples are mentioned in this Consumers Union list, but local varieties known as "Harvest apples" are fine, though usually they won't keep. Most apples improve by long keeping, in fact "a last year's apple" is demanded by epicurean orchardists and readers of Proust. To our mind there's no better Sunday night supper than snow apples and a jug of sparkling hard cider, each trying to outsnap the other. Snow apples are so small and delicate they're often eaten seeds and all, so "there ain't gonna be no core." And for jelly, the ugly little crabapple that's much too puckery to gnaw raw, is a natural. Any other apple sliced into beer soup, or made into a Waldorf Salad with celery and nuts, is something to brag about. Apricots. Best, of course, are tree-ripened. Since they're perishable they're best buys only near where grown in California, Oregon and Washington. The Persians call them "sun eggs," which shows they shouldn't be picked green. Avocados. (Alligator Pears.) Called "tropical salad fruit." There are many different kinds, those from California trade-named G?/avo (California and avocado) those from Florida, Flavocado. Rough, thick-skinned ones are fullest of flavor. When ready for eating they must be soft as butter (in fact, they are fresh vegetable butter). Don't buy them when bruised or discolored. Cheapest in fall and winter. Bananas. Ripe bananas are best and can be bought advantageously when dealers can't keep a big ripe stock over the week-end. Must be fat and juicy with brown spots proving they're sugary. Don't be afraid of black skins if the fruit is sound inside. Bananas, by the way, are the only fruit we know that actually improves by being picked green. Yet we remember the rarer flavor of the naturally ripened fruit in Brazil. Berries. Must be fresh and plump. Red raspberries mildew quickly and should be used at once. Always have the seller turn out the berries so you can see what's on the bottom, and remember that "short pack" cheating in measure is common sharp practice.
- 44. THE PICK OF THE PUSHCART 1 75 Never buy "a pig in a poke." This business of buying sight-unseen, for which we fall too easily in our hurried, harried struggle for any life at all, only profits the seller. Berries with their caps on are likely to be green and juiceless. Make sure they're plump and juicy, but not wet and soggy, for then you'll waste more than you'll use. Your eye quickly learns to detect ripeness by full color and it's always a good idea to taste a berry or two if the seller is willing; and if they're any good at all, he is, for it pays him to let you prove "the pudding" by eating thereof. Blueberries are a better buy than drier, too seedy huckleberries. Cherries. Sour cherries for cooking, sweet ones for eating. They must be bright and shiny, fat, yet firm. Look out for worms and dull- skinned ones, which show they're too long off the tree. Cranberries. Good ones must be crisp and tender; by pressing one between thumb and forefinger you can hear the hearty snap. Those out of cold storage are apt to be leathery, shrivelled, soggy, or all three. As with all berries, they should be bought as fresh from the bush as possible. For making jelly and most preserves, fruit shouldn't have been picked earlier than the day before it's used. The fresher the better; and the most pectin, when it is a bit under-ripe. Pectin is the jelly-making constituent and it can't be duplicated by manufactured pectins, no matter what the siren ads say. Figs. Come in all colors of the rainbow and all sizes, each as good as the other in its way, but every one of them must be mellowly ripe to be worth a damn. They sour and ferment quickly, so watch your step! Many are being shipped from the blooming deserts of Arizona and New Mexico and we buy them in Paddy's market in prime condition for a cent apiece. With cream, they make as exceptional dessert as peaches and cream, at about a nickel per eater. Grapefruit. Seedless ones are a better buy than those full of big slippery buck teeth that sometimes take up half the space inside. But the best way to recognize your money's worth is by weight in relation to size, the heavier, the more juice for your money. Russet grapefruit looks rusty and on that account is sold cheaper in some places, but the flavor is just the
- 45. same, or even better — as with brown and white eggs, white eggs always sell at a premium in
- 46. 176 MOST-FOR-YOUR-MONEY COOK BOOK New York while in the middle West brown ones may cost a little more. But both have the same nutritive value, which Consumers Union says is also true of Grade A and Grade B milk. Our pick of grapefruit is the pink-fleshed seedless one from Texas. It*s delicately yet fully flavored, pretty to look at and shouldn't cost more than the pallid white kind. Grapes. There are so many kinds of grapes, both domestic and imported, that only general rules can be given. They must be plump, heavy, fresh looking, fully colored and stuck firmly to the stem. If you gently shake a bunch and the grapes rattle off* like hail it's no good — and the seller will curse you, besides. But what are you going to do ? It's always wise to sample one, for some of fine quality when kept packed in sawdust too long taste like old corks. One of the best seasonal buys is the small seedless California kind that's been burbanked. It's easy to eat and sweet as Tokay wine. Lemons. Thin skin and golden color for plenty of juice. Some of the small thick- skinned ones are a bad buy; they don't give as much juice as a lime and are little better value than the old wooden nutmeg. Lemons are cheap in California, which produces the bulk of our supply, but in the East they're two or three for a dime even at the height of their season. At the same time, however, they're peddled on the pushcarts, just as good, even better quality, at five or six for a dime, but here you have to watch out for tiny ones, no bigger than pullet eggs, and both of them look fair-sized if there isn't a regular lemon or cackleberry handy to compare them with. To keep lemons fresh drop them in a jar of water and take them out as needed. That keeps them juicy and prevents the skin from mummyfying. Limes. Dark green ones are prime and the best of these are the big ones called Persians. But unless one is addicted to the piquant flavor of limes in sloe gin rickeys and such, we consider lemons, even small ones, a better buy. MusKMELLONS OR CANTALOUPES. Since most melons for market are picked green in the condition known as "half- slip," they're usually pretty punk and it's wise to ask for a slice of the same run of melons before buying. We've wasted so much money on these tasteless muskies that we're almost afraid to look one in the
- 47. squinting half-slip eye. They're tempting to view, and sometimes wink at
- 48. THE PICK OF THE PUSHCART 1 77 you, but you simply can't tell anything about the taste from outside appearance, except by smelling the stem end and that takes a really educated smeller. Finely-netted Rockyfords are fairly safe. Ripeness can be told by pressing; if it's firm and elastic that part's all right. With honey-dews and honey balls, however, it's wise to keep them a long time until you're dead sure they're dead ripe; otherwise you'll just have to throw them away. No melon should be kept in a damp place — that takes the flavor out. Nectarines. This cross of apricot and peach is pretty to look at, but usually not as mellow and full-flavored as either a peach or an apricot just by itself. In fact, if you bite into a nectarine with your eyes shut you'll say it's just a hard, tasteless peach. We've never had them fully-ripened from the tree, so this goes only for the kind picked green and shipped all the way across the continent, bumpety-bumpety. Oranges. See 21 Ways to Eat i Orange, The best buys in this line are "grove or orchard run" trucked direct to town markets; they may not look as fancy because they aren't washed, waxed or polished, but they taste every bit as good, nay, even better. In New York's pushcart market known as "Paddy's" we always find bargains in the tangerine types, big King and mandarin oranges and the smaller satsumas, sometimes bursting in taste, and we pick them for tightness of skin in relation to weight. The light ones with the loose wrinkled skins simply aren't worth buying. Get the cannon-ball kind which go two or three for a nickel in season, after the bon voyage baskets have been filled with all they can hold, at a dime apiece. Peaches. A peach may look pretty and yet taste pretty punk; so the only sure test is to eat a slice. As with apples or tomatoes, when buying by the basket, ask to see what's at the bottom and if there aren't too many tiny, green, gnarly, wormy and rotten ones, they're worth buying if the price is fair. Reject spotted and bruised ones — too much waste. Pears. Weight according to size is the best test, but as with peaches, it's wise to taste before buying. You can always buy one to test, and that may save you the price of a whole worthless dozen. Pineapples. The only test for ripeness is to pluck a green blade
- 49. 178 MOST-FOR-YOUR-MONEY COOK BOOK from the center of the bunch that thrusts out at the top. But smell also comes into play, as with melons. Like figs, pineapples quickly ferment, so don't buy any with soft spots or squashy bottoms. As a matter of fact, although we always prefer any fresh fruit to canned, the Hawaiian canned slices and spears have an advantage over the fresh fruit in that they are picked fully ripe. We like the snap of a fine, fresh pine, but when we stop to figure that it costs three or four times as much as the canned article we pass it up. In general, a good canned brand is more dependable and brings with it a whole lot of appetizing juice that comes in handy for drinks and sauces. So, our mind is made up that canned pineapple is the very best fruit buy the year 'round. You're certain of quality with absolutely no waste. And canned slices or spears are a much better buy than just the juice, because you get both the fruit and a richer juice. Plums and Prunes. Prunes are a species of plum best suited for drying, but when fresh they're as fine as any plum or apricot. In fact Damsons are shipped from Washington under the name "Italian prunes." A grandmother of ours had a prune tree on her front lawn in Lansing, Michigan, and when the fruit was ripe we couldn't get enough of it. But since most of us haven't a yard, let alone a prune tree, we'll have to use what plums we can get and these are usually good value, if mellowly ripe but not squashy. Since the flavor varies widely with different kinds, it's always wise to sample one before buying a whole basket, for a juiceless plum can be as unpalatable as a wrinkled sun-dried prune. Try peeled green-gages just plain for Sunday night supper, or plum cake with tea. Pomegranates. One of the best pushcart buys we know is this classical food of Prosperpine, now grown abundantly in the West and shipped everywhere. The heaviest, biggest sunset- glowing ones are most-for-your-money and that money is usually a nickel apiece, or less, because only foreigners, Greeks and Italians especially, really appreciate this exceptional fruit. We Browns like the bittersweet tartness, yet often we leave them sitting around till they begin to wizen and dry, because we like their exotic decorative appearance even more than that of Mexican ornamental gourds —
- 50. probably because you can actually eat them after enjoying their colorful warmth. And another reason we don't rush to eat them is
- 51. THE PICK OF THE PUSHCART I79 because they're too hard to get at and then nibble, pip by pip. That's one way to eat them, but the worst. It's better to collect a mouthful at a time; but the easiest way is to cut a wide hole in the end, being careful to get rid of all bitter skin and then squeeze gently and suck as you would an orange. Very gentle squeezing is necessary, because if you bruise the skin or any of the million membranes, fibres, cores, or even the seeds themselves, you get a mouthpuckering dose. The juice makes a good drink by itself, with ice and plain or fizz water — we call it pomegranatade. You can also put a kick in a fruit salad by squeezing some tart pomegranate juice over it, or add a refreshing acid touch to such a drink as a gin rickey. But time and trouble must be taken to get only the juice of the pips and none of the skin, pith or seeds. The Syrians remove the pips in clumps, cut away the bitter cores and stew with sugar, flavoring with rosewater at the finish. Quinces. To be worth buying, quinces must be firm, but not hard; and yellowish, but not green. Otherwise they're too puckery for eating. The immature kind aren't even fit for canning, and bruised or wormy ones aren't worth carrying home. Strawberries. Wild strawberries are most flavorsome, but they seldom come to market; so, among the tame varieties, pick fresh, bright-colored ones, not for bigness, but for soundness of meat, because big ones can be almost as pithy as overripe radishes. Unlike other berries, it's safest to buy strawberries with their caps on; capless ones are likely to be mushy and worthless. Again, the ojily sure test is to taste one and, of course, the box should be dumped out into the dealer's palms to see just what you're getting at the bottom. Watermelons. Don't buy a watermelon without having it plugged. If the melon is big, the rind thin and the flesh deep red and juicy you'll find it good fruit value for the money. The bigger the melon the bigger its heart. And since watermelons that get to the city are high in price from their first appearance on Fourth of July all the way to Labor Day, don't waste the rind; preserve or crystallize it. The seeds when dried make good eating, too; so no part of a watermelon need be wasted. Not even
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