mth263 lecture 2

MTH 263 Probability and Random Variables

Lecture 2, Chapter 2Dr. Sobia Baig

Electrical Engineering DepartmentCOMSATS Institute of Information Technology, Lahore

ContentsContents

• RANDOM EXPERIMENTSRANDOM 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 consistWe consider sequential random experiments that consist of performing a sequence of simple random subexperiments.


RANDOM EXPERIMENTSRANDOM EXPERIMENTS

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


SPECIFYING RANDOM EXPERIMENTSSPECIFYING RANDOM EXPERIMENTS

• A random experiment is specified by statingA 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 ExperimentsExamples of Random Experiments


Sample SpaceSample Space

• The sample space S of a random experiment isThe 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 f f d i• Each performance of a random experiment can then be viewed as the selection at random f i l i ( ) f Sof a single point (outcome) from S.


Specifying Sample SpaceSpecifying Sample Space

• The sample space S can be specifiedThe 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.


SetsSets

• 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 2.1Sample Space of Example 2.1


Sample Space ClassificationSample Space Classification

• A sample space can be p p– Finite– countably infinite

t bl i fi it– uncountably infinite• We call S a discrete sample space if S iscountable; that is, its outcomes can be put intocountable; that is, its outcomes can be put intoone‐to‐one correspondence with the positiveintegers.W ll S i l if S i• We call S a continuous sample space if S is not countable. Experiments and have finite discrete sample spaces.p p


Sample Space Classification‐ExamplesSample Space Classification Examples


EventsEvents

• 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 ibl ll t hi h t i• the impossible or null event, which contains no outcomes and hence never occurs Ø

• An event from a discrete sample space that consists of p p fa single outcome is called an elementary event.

• Which event in previous slide is Elementary?


Example of EventsExample of Events


Set Theory & Venn Diagrams

Concepts from Set Theory.Co cepts o Set eo y

• A set is a collection of objects and will be denoted by capital letters

• We define U as the i l t th tuniversal set that

consists of all possible objects of interest in a j fgiven setting or application.

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

16

Set OperationsSet Operations


THE AXIOMS OF PROBABILITYTHE AXIOMS OF PROBABILITY

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


Discrete Sample SpacesDiscrete 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 ,…,a } P(B) P{a , 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 SpacesContinuous Sample Spaces

• Continuous sample spaces arise in experiments in which the t b th t ti f loutcomes are numbers that can assume a continuum of values, so

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].


ExampleExample

• Consider Experiment where we picked twoConsider 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 inreasonable 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 SpaceContinuous Sample Space


SummarySummary

• A probability model is specified by identifying the sample space S, th t l f i t t d i iti l b bilit i tthe event class of interest, and an initial probability assignment, a “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

i f h i f l Wh S i iconsist of the union of elementary events.When S is continuous, 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 ti fi d b th b biliti f tsatisfied by the probabilities of events.

• The corollaries that follow from the axioms provide rules for computing the probabilities of events in terms of the probabilitiescomputing 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 theIf 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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