CS723 - Probability and Stochastic Processes

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CS723 - Probability and Stochastic Processes

Lecture No. 27

In Previous Lectures Transformation of random variables Derived PDF of random variable (s) from the PDF of underlying random variable (s) Examples: Y = g(X) Z = g(X,Y) U = g(X,Y) & V = h(X,Y) fY(y) / fUV(u,v) were found via CDF and via direct processing of fX(x) / fXY(x,y) fZ(z) was obtained from FZ(z) that was computed from fXY(x,y)

fY(y) Directly from fX(x)

fY(y) Directly from fX(x)

fuv(U,V) from fXy(x,y)

fuv(U,V) from fXy(x,y)

fZ(z) from fXy(x,y)

E[Y=g(X)] from fX(x)

E[Y=g (X)] from fX(x)

E[Y=g(X)] from fX(x) Some examples of simple transformations

Moments from fXY(x,y) Direct use of fXY(x,y), the joint PDF of X and Y RXY and ρXY HAVE to use joint PDF fXY(x,y)

Moments of U&V from fXY(x,y) E[g(x,y)] using joint PDF fXY(x,y)

Moments of U&V from fXY(x,y) E[g(x,y)] using joint PDF fXY(x,y)

Moments of U&V from fXY(x,y)

Moments of U&V from fXY(x,y) Similarly, Examples of linear transformations functions g( . , . ) and h( . , . ) To be continued …