Mth263 lecture 2

MTH 263
Probability and Random Variables
Lecture 2, Chapter 2
Dr. Sobia Baig
Electrical Engineering Department
COMSATS Institute of Information Technology, Lahore

Contents
• RANDOM EXPERIMENTS
RANDOM EXPERIMENTS
– Examples
• THE AXIOMS OF PROBABILITY
THE AXIOMS OF PROBABILITY
– Examples
• Sample Space
– Discrete
– Continuous

Probability and Random Variables, Lecture 2, by Dr. Sobia Baig 2

Basic Concepts of Probability Theory

• set theory is used to specify the sample space and the
events of a random experiment
• the axioms of probability specify rules for computing the
probabilities of events
• the notion of conditional probability allows us to determine
how partial information about the outcome of an
experiment affects the probabilities of events
• Conditional probability also allows us to formulate the
notion of “independence” of events and of experiments.
• We consider “sequential” random experiments that consist
We consider sequential random experiments that consist
of performing a sequence of simple random
subexperiments.


RANDOM EXPERIMENTS
RANDOM EXPERIMENTS
• A random experiment is an experiment in
A random experiment is an experiment in
which the outcome varies in an unpredictable
fashion when the experiment is repeated
fashion when the experiment is repeated
under the same conditions.


SPECIFYING RANDOM EXPERIMENTS
SPECIFYING RANDOM EXPERIMENTS
• A random experiment is specified by stating
A random experiment is specified by stating
– an experimental procedure and
– a set of one or more measurements or
a set of one or more measurements or
observations


Examples of Random Experiments


Sample Space
Sample Space
• The sample space S of a random experiment is
The sample space S of a random experiment is
defined as the set of all possible outcomes.
• We will denote an outcome of an experiment
We will denote an outcome of an experiment
by ζ where ζ is an element or point in S.
• E h
Each performance of a random experiment
f f d i
can then be viewed as the selection at random
of a single point (outcome) from S.
f i l i ( )f S


Specifying Sample Space
Specifying Sample Space
• The sample space S can be specified
The sample space S can be specified
compactly by using set notation.
• It can be visualized by
It can be visualized by
– drawing tables
– Di
Diagrams
– intervals of the real line
– regions of the plane.


Sets
• There are two basic ways to specify a set:
There are two basic ways to specify a set:
1. List all the elements, separated by commas,
inside a pair of braces:
inside a pair of braces:
A={1,2,3}
2. Give a property that specifies the elements of
the set:
• A = {x: x is an integer such that 0 ≤x ≤ 36}


Sample Space of Example 2.1
Sample Space of Example 2.1


Sample Space Classification
Sample Space Classification
• A sample space can be
p p
– Finite
– countably infinite
– uncountably i fi it
t bl infinite
• We call S a discrete sample space if S is
countable; that is, its outcomes can be put into
one‐to‐one correspondence with the positive
integers.
• W
We call S a continuous sample space if S is not
ll S i l if S i
countable. Experiments and have finite discrete
p p
sample spaces.

Sample Space Classification Examples
Sample Space Classification‐Examples


Events
• Events are subsets of Sample Space
• We may not be interested in a single outcome, instead
in occurrence of an event
• An event of special interest is the certain event S
An event of special interest is the certain event, S,
which consists of all outcomes and hence always
occurs.
• th i
the impossible or null event, which contains no
ibl ll t hi h t i
outcomes and hence never occurs Ø
• An event from a discrete sample space that consists of
p p f
a single outcome is called an elementary event.
• Which event in previous slide is Elementary?


Example of Events
Example of Events


Set Theory & Venn Diagrams

Co cepts o Set eo y
Concepts from Set Theory.
• A set is a collection of
objects and will be
denoted by capital letters
• We define U as the
universal set that
i l t th t
consists of all possible
objects of interest in a
j f
given setting or
application.

Probability and Random Variables, Lecture
16
2, by Dr. Sobia Baig

Set Operations
Set Operations


• Probabilities are numbers assigned to events
Probabilities are numbers assigned to events
that indicate how “likely” it is that the events
will occur when an experiment is performed
will occur when an experiment is performed


Discrete Sample Spaces
Discrete Sample Spaces
• Discrete vs. Continuous Samples
• the probability law for an experiment with a countable
sample space can be specified by giving the probabilities of
the elementary events
• Given a finite sample space S = {a1, a2,…,an} and all distinct
elements are disjoint, then the probability of an event
• B = {a1’, a2’,…,am’} P(B) = P{a1’, a2’,…,am’} = P(a1’) + …P(
B {a , a } P(B) P{a , a } P(a ) …P(
am’) ,
• and if an event is D = {b1’, b2’,… } P(D) = P(b1’) + P(b2’)…
• Equally likely outcomes: Then the probability of an event is
Equally likely outcomes: Then the probability of an event is
equal to the number of outcomes in the event divided by
the total number of outcomes in S


Continuous Sample Spaces
Continuous Sample Spaces
• Continuous sample spaces arise in experiments in which the
outcomes are numbers that can assume a continuum of values, so
t b th t ti f l
we let the sample space S be the entire real line R (or some interval
of the real line).

• Consider the random experiment “pick a number x at random
between zero and one.

• The sample space S for this experiment is the unit interval [0, 1],
which is uncountably infinite.

• If we suppose that all the outcomes S are equally likely to be
selected, then we would guess that the probability that the
outcome is in the interval [0, 1/2] is the same as the probability that
the outcome is in the interval [1/2, 1].
the outcome is in the interval [1/2 1]


Example
• Consider Experiment where we picked two
Consider Experiment where we picked two
numbers x and y at random between zero and
one.
• The sample space is then the unit square.
• If we suppose that all pairs of numbers in the unit
If we suppose that all pairs of numbers in the unit
square are equally likely to be selected, then it is
reasonable to use a probability assignment in
reasonable to use a probability assignment in
which the probability of any region R inside the
unit square is equal to the area of R.
q q

Continuous Sample Space


Summary
• A probability model is specified by identifying the sample space S,
the event class of interest, and an initial probability assignment, a
th t l fi t t d i iti l b bilit i t
“probability law,” from which the probability of all events can be
computed.

• The sample space S specifies the set of all possible outcomes. If it has
a finite or countable number of elements, S is discrete; S is
continuous otherwise.

• Events are subsets of S that result from specifying conditions that are
of interest in the particular experiment.When S is discrete, events
consist of the union of elementary events.When S i
i f h i f l Wh S is continuous,
i
events consist of the union or intersection of intervals in the real
line.


Summary …
Summary …
• The axioms of probability specify a set of properties that must be
satisfied by the probabilities of events.
ti fi d b th b biliti f t

• The corollaries that follow from the axioms provide rules for
computing the probabilities of events in terms of the probabilities
computing the probabilities of events in terms of the probabilities
of other related events.

• An initial probability assignment that specifies the probability of
• An initial probability assignment that specifies the probability of
certain events must be determined as part of the modeling.

• If S is discrete it suffices to specify the probabilities of the
If S is discrete, it suffices to specify the probabilities of the
elementary events. If S is continuous, it suffices to specify the
probabilities of intervals or of semi‐infinite intervals.


Mth263 lecture 2

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