CS723 - Probability and Stochastic Processes

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Presentation transcript:

CS723 - Probability and Stochastic Processes

Lecture No. 22

In Previous Lectures A scalar-valued random variable that is a function of two RV’s Examples: Z = X+Y and Z = Y – X The example involved two televisions, the sum of their service times, and difference of their service times PDF and CDF of Z from the joint PDF of X & Y Answer questions about Z directly by processing fZ(z)

Function of Two RV’s Ratio of service times of two TV’s Z = Y / X

Evaluation of FZ(z)

CDF of Z = Y / X FZ(z) = z / (z + 2)

PDF of Z = Y / X fZ(z) = 1/(2+z) – z/(2+z)2

Events for Z = Y / X Pr( 0.5 < Z ) & Pr( Z > 2 )

Function of Two RV’s Multiplication of service times Z = XY

Evaluation of FZ(z)

Evaluation of FZ(z) The double integral to get FZ(z) is given by

Function of Two RV’s Max of service times of two TV’s Z = max(X,Y)

Evaluation of FZ(z) The parameter z only takes on positive values The area of integration is always a square Due to independence, the probability of this square is easy to find

CDF of Z = max(X,Y)

PDF of Z = max(X,Y)

Events for Z = max(X,Y) Pr( Z < 3 ) & Pr( Z > 30)

Function of Two RV’s Min of service times of two TV’s Z = min(X,Y)

Evaluation of FZ(z) The parameter z only takes on positive values The area of integration is always a square Due to independence, the probability of this square is easy to find

CDF of Z = min(X,Y)

Function of Two RV’s Distance from origin to (x,y) Z = (X2 + Y2)1/2