CS723 - Probability and Stochastic Processes

Lecture No. 16

In Previous Lectures Analysis of continuous random variables Uniformly distributed continuous random variable with a flat PDF Exponentially distributed continuous random with fX(x) = λ e- λx Erlang type continuous random variable with fX(x) = λ2 x e- λx Gaussian continuous random variable with fX(x) = sqrt(λ/π) e- λx2

Gaussian Random Variable

Drop of a Ball A thin horizontal line on the wall is to be hit by a tennis ball

Expected Value of a Cont. RV Defined analogues to expected value of a discrete random variable E(X) = X =  x fX(x) dx Expected value of a uniform random variable defined over [a,b] is (a+b)/2 Expected value of an exponential random with PDF λ e-λx is 1/λ Expected value of Erlang random variable with PDF λ2 x e-λx is 2/λ

Graphical Interpretation

Mean of Transformed RV The expected value of Y=g(X) can be directly computed from PDF of X E(Y) =  Y fY(y) dy =  g(x) fX(x) dx A linear transformation Y = g(X) directly transform the value of the mean E(Y) = Y = g(X) Non-linear transformations require re-evaluation of Y =  g(x) fX(x) dx If Y = g(X) = X - X , then E(Y) = 0

Height Distribution of Humans A large population of adult males and females follow following PDF’s

Height Distribution of Humans A few problems related to the Gaussian height distributions of males & female

Higher Moments Defined analogues to expected value of a discrete random variable E(X) = X =  x2 fX(x) dx Expected value of a uniform random variable defined over [-a,a] is (a)2/3 Expected value of an exponential random with PDF λ e-λx is 1/λ2 Expected value of Erlang random variable with PDF λ2 x e-λx is 2/λ2