Lecture 16: Gaussian Random Variables & Expected Values, Slides of Probability and Stochastic Processes

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CS723 - ProbabilityCS723 - ProbabilityandandStochastic ProcessesStochastic Processes

• Lecture No. 16Lecture No.

Gaussian Random VariableGaussian Random Variable PDF's of a family of Gaussian random variables 1 0.750.5 0.25 0 -2^ -^

0 1

2 (^1) 0.5 0 CDF's of PDF's given above

Drop of a BallDrop of a BallA thin horizontal line on the wall is to be

hit

A thin horizontal line on the wall is to beby a tennis ball

hit

by a tennis ball

Graphical InterpretationGraphical Interpretation 0 1

2 3

4 Expected values of exponetial and Erlang random variables (^21) 0.5 (^0) 0.5 0

Mean of Transformed RVMean of Transformed RV• The expected value of Y=g(X)

can^ be directly

computed from PDF of XE(Y) =^ ^ Y f^ (y) dy =Y^

^ g(x) f^ (x) dxX^

-^ A linear transformation Y = g(X) directly

transform

the value of the meanE(Y) =^  = g(Y^

 )X

-^ Non-linear transformations requirere-evaluation of^  =Y^

^ g(x) f^ (x) dxX^

-^ If Y = g(X) = X -^  ,^ X^

then E(Y) = 0

Height Distribution of HumansHeight Distribution of Humans A few problems related to the GaussianA few problems related to the Gaussianheight distributions of males & femaleheight distributions of males & female

Higher MomentsHigher Moments • Defined analogues to expected value of

a discrete random variableE(X) =^ =^ ^ x2 f^ (x) dxX^ X • Expected value of a uniform randomvariable defined over [-a,a] is (a)2/3 • Expected value of an exponential random- λ x^ with PDF^ λ^ e^ is 1/

λ^2

-^ Expected value of Erlang random variable^2 with PDF^ λ x e - λ x^ is 2/ λ^2