STATISTICS and PROBABILITY.pptx file for Stats
- 1. STATISTICS and PROBABILITY RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
- 2. INTRODUCTION The module is intended for you to illustrate a random variable, distinguish between a discrete and a continuous random variable and find the possible values of the variable.
- 3. OBJECTIVES After going through this lesson, you are expected to illustrate a probability distribution for a discrete random variable and its properties.
- 4. Illustrating Random Variables and Distinguishing Between a Discrete and Continuous “random” - often read and heard from people of different walks of life. Like, “the distribution of relief goods is randomly chosen in our barangay”
- 5. Illustrating Random Variables and Distinguishing Between a Discrete and Continuous “There is a random SWAB testing which will be conducted by our local health unit on Saturday”. But does it tell about a real random event? No, this is a decision that was made on the basis of other variables as desire and the lack of satisfaction with other options. The word random has a different meaning in the field of statistics. It is random when it varies by chance.
- 6. Activity 1: Tossing a coin As you can see in a one-peso coin, it has Dr. Jose P. Rizal on one side, which we call it as Head (H), and the other side as the Tail (T). Toss your one peso coin three times and record in your notebook the results of the three tosses. In order to write the result easily, use letter H for the heads and letter T for the tails. If the results of your three tosses are head, head, head, then you will write HHH on your notebook. Example 1: How many heads when we toss 3 coins? Continue tossing your coin and record the time. If possible, use mobile phone timer and record up to the last minutes.
- 7. Activity 1: Tossing a coin Let say in a minute, how many times the heads and tails appeared. Then record all the possible answers on your notebook. Write all eight possible outcomes. You can do this systematically so that you do not get confused later on. You have noticed that, there are 0 head, 1 head, 2 heads, or 3 heads. Thus, the sample space is equal to 0,1,2,3. Then this time the results or outcomes are NOT entirely equally likely. The three coins land in eight possible ways: X= Number of Head Looking at the table below, we see just 1 case of Three Heads, but 3 cases of Two Heads, 3 cases of One Head, and 1 case of Zero Head.
- 8. Looking at the table below, we see just 1 case of Three Heads, but 3 cases of Two Heads, 3 cases of One Head, and 1 case of Zero Head.
- 9. So, P(X=3) = 1/8 P(X=2) = 3/8 P(X=1) = 3/8 P(X=1) = 1/8 discrete variable
- 10. A random variable is called discrete if it has either a finite or a countable number of possible values. Thus, a discrete random variable X has possible values 1, , ,…. 𝑥 𝒙𝟐 𝒙𝟑 A random variable is called continuous if its possible values contain a whole interval of numbers.
- 11. From another source, a random variable is a numerical quantity that is generated by a random experiment. (Malate, 2018). We will denote random variables by capital letters, such as X or Z, and the actual values that they can take by lowercase, such as x and z.
- 12. Examples of random variables
- 13. About random variables a. That a random variable must take exactly one value for each random outcome. b. That random variables are conceptually different from the mathematical variables that they have met before in math classes. A random variable is linked to observations in the real world, where uncertainty it involved. c. Random variables are used to model outcomes of random processes that cannot be predicted deterministically in advance (the range of numerical outcomes may, however be viewed).
- 14. Definition of Terms A random variable is a numerical quantity that is generated by a random experiment. (Malate, 2018) is discrete if it has a finite or countable number of possible outcomes that can be listed. is called discrete if it has either a finite or a countable number of possible values. A random variable is called continuous if its possible values contain a whole interval of numbers. (Malate, 2018) is continuous if it has an uncountable number or possible outcomes, represented by the intervals on a number line. (Course Hero n.d.)
- 15. Definition of Terms STATISTICS – a branch of Mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data - a collection of quantitative data PROBABILITY – the chance that a given event will occur - the ratio of the number of outcomes in an exhaustive set of equally likely outcomes that produce a given event to the total number of possible outcomes
- 16. Example: Discrete Random Variable 1. Number of heads in 4 flips of a coin (possible outcomes are 0, 1, 2, 3, 4) 2. Number of classes missed from March 2020 to December 2020 3. The number of siblings a person has 4. The number of Covid-19 cases in Negros Oriental in 2020 5. The number of students involve in Online classes in Neg. Or. Division during this pandemic time
- 17. Example: Continuous Random Variables 1. Heights of students in a class 2. Time to finish a module 3. Hours spent exercising 4. Distance travelled from Dumaguete to Pamplona A continuous variable is a value that is being acquired by measuring.
- 18. ACTIVITY 2 Complete the following table. The first one is done for you.
- 19. Enrichment Activities Activity 3 In tossing a coin four times, how many outcomes correspond to each value of the random variable? What if the coin would be tossed five times? six times? seven times? eight times? Try to relate the outcomes to the numbers in Pascal’s triangle.
- 20. For tossing the coin four times, there will be five possible values, 0, 1, 2, 3, 4, with 1, 4, 6, 4, 1 outcomes, respectively. For five coins there are six possible values, 0, 1, 2, 3, 4, and 5, with 1, 5, 10, 10, 5, 1 outcomes, respectively. In general, for n tosses of a coin, there are n+1 possible values, 0 ,1, 2, 3,…, n. If k is a possible value, then there are = ( )= ! !( − )! outcomes 𝑐𝐶𝑥 𝑛 𝑘 𝑛 𝑘 𝑛 𝑘 associated with x. (Abacea 2016)
- 21. Independent Activity 4 Look back and reflect. 1. How do you determine the values of a random variable? 2. How do you know whether a random variable is continuous or discrete? 3. What is the difference between the two types of random variables?
- 24. ASSESSMENT
Editor's Notes
- #12: The four examples in the table above are random variables. In the second example, the three dots indicate that every counting number is a possible value for X. The set of possible values is infinite, but is still at least countable, in the sense that all possible values can be listed one after another. In the last two examples, by way of contrast, the possible values cannot be individually listed, but take up a whole interval of numbers. In the fourth example, since the light bulb could conceivably continue to light indefinitely, there is no natural greatest value for its lifetime, so we simply place the symbol ∞ for infinity as the right endpoint of the interval of possible values. (Saylor Academy 2012)