Probability and Stochastic Processes - Bucknell …kozick/elec47102/prob_ch2.pdfProbability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers Chapter

Probability and Stochastic ProcessesA FriendlyIntroductionfor ElectricalandComputerEngineers

Chapter 2 Viewgraphs

1

Random Variables

Experiment:Procedure+ Observations

Observationis anoutcome

Assigna numberto eachoutcome:Randomvariable

2

Random Variables

Threewaysto geta rv:

Therv is theobservation

Therv is a functionof theobservation

Therv is a functionof a rv

3

Discrete Random Variables

� rangeof (setof possiblevalues)

is discreteis � is countable

Discreterv hasPMF

�

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PMF Properties

�� � � �

For anevent � ,

� ��

5

Bernoulli RV

Getthephonenumberof a randomstudent.Let if thelastdigit is

even.Otherwise,let .

�otherwise

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Binomial RV

Test circuits,eachcircuit is rejectedwith probability independentof

othertests.

no. of rejects

is thenumberof successesin trials:

�

� � � � �

otherwise

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Geometric RV

Circuit rejectedwith prob . is thenumberof testsup to andincludingthe

first reject.

� � � � � �� � � � � � �� � � � � � ��

� � � � � �� � �

Fromthetree, , ,

� �otherwise

8

Geometric:

� �

otherwise

0 10 200

0.1

0.2

y

PY(y

)

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Pascal RV

No. of tests, , neededto find rejects.

rejectsin tests

successon attempt

Events and areindependent

10

Pascal continued

and is binomial:

succ.in trials

� � � � � � � �

�

� �� � � � �

otherwise

11

Pascal: ,

�

� �� � � �

otherwise

0 10 20 30 400

0.05

0.1

l

PL(l

)

12

Summary

Bernoulli No. of succ.ononetrial

Binomial No. of succon trials

Geometric No. of trialsuntil first succ.

Pascal No. of trials until succ

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Poisson rv

Countsarrivals of something.

Arrival rate , interval time .

With ,�

� �

otherwise

14

Poisson:

�

� � ��

otherwise

0 2 40

0.5

1

j

PJ(j

)

15

Poisson:

�

� �

otherwise

0 5 10 150

0.1

0.2

j

PJ(j

)

16

Cumulative Distribution Functions

Thecumulative distribution function (CDF) of randomvariable is

�

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CDF Example

−1 0 1 2 30

0.5

1

r

PR(r

)

−1 0 1 2 30

0.5

1

r

FR(r

)

At thediscontinuities and , � is theuppervalues.(right

handlimit)

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CDF Properties

For any discreterv , range � � � satisfying � � ,� and �

For all , � �

For � � andsmall ,� � � � � �

� � � for � � � �

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Expected Value

Theexpectedvalueof is

�� � �

�

Also calledtheaverageof .

20

Average vs.

Averageof samples: � ����� �

Each � . If each � occurs � times,�

� � ��

� � ��

� � ��

21

Derived Random Variables

Eachsamplevalue of a derived rv is a function of a sample

value of a rv .

Notation:

ExperimentalProcedure

1. Performexperiment,observe outcome .

2. Find , thevalueof

3. Calculate

22

PMF of

� ! � � �� ��

23

Problem 2.6.5

Sourcetransmitsdatapacketsto receiver.

If rec’d packet is error-free,rec’vr sendsbackACK, otherwiseNAK sent.

For eachNAK, thepacket is resent.

Eachpacket transmissionis independentlycorruptedwith prob .

Find thePMF of , no. of timesapacket is sent

Eachpacket takes1 msecto transmit.Sourcewaits1 msecto receive

ACK. equalthetime req’d until thepacket is receivedOK. Whatis

" ?

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Expected value of

Thm: Givenrv with PMF � , theexpectedvalueof , is

� � �

�

Example: :

� � �

�

25

Variance and Std Deviation

Variance: � �� � �

� � �

Variancemeasuresspreadof PMF

StandardDeviation: �

Unitsof � arethesameas .

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Properties of the variance

If , .

If , � .

27

Conditional PMF of given

Given , with ,

� �

Two kindsof conditioning.

28

Conditional PMFs - version 1

Probabilitymodeltellsus � � # for possible � .

Example:In the th monthof theyear, thenumberof cars crossingthe

GW bridgeis Poissonwith param � .

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Conditional PMFs - version 2

is aneventdefinedin termsof .

is a subsetof � suchthatfor each � , either or .� �

$ � � � �$ % � &

otherwise

30

Conditional PMF Example

Example: is geometricwith . Whatis theconditionalPMF of

givenevent that ?

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Conditional Expectations

Replace � with � �

� � �� � �

�

32