Chapter 4 Joint Distribution & Function of rV. Joint Discrete Distribution Definition.

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Presentation transcript:

Chapter 4 Joint Distribution & Function of rV

Joint Discrete Distribution Definition

Joint discrete CDF

Example for joint distributions Consider the following table: Using the table, we have Y=0 Y=3 Y=4 X=5 1/7 1/7 1/7 3/7 X=8 3/7 0 1/7 4/7 4/7 1/7 2/7 pXpX pYpY

The Marginal PDF

Example : Air Conditioner Maintenance –A company that services air conditioner units in residences and office blocks is interested in how to schedule its technicians in the most efficient manner –The random variable X, taking the values 1,2,3 and 4, is the service time in hours –The random variable Y, taking the values 1,2 and 3, is the number of air conditioner units

Expected Values for Jointly Distributed Random Variables Let X and Y be discrete random variables with joint probability density function p(x, y). Let the sets of values of X and Y be A and B, resp. We define E(X) and E(Y) as For the random variables X and Y from the previous slide example,

Joint p.d.f Joint cdf Y= number of units X=service time Find E[X] and E[Y] !!

Previously Example –Marginal p.d.f of X –Marginal p.d.f of Y

Joint continuous distribution

Th

Ex a. Then find the joint CDF !

The Marginal Continuous PDF FindFrom previously example

Ex

Independent rV

Conditional pdf

Joint MGF

Group Discuss the exercise bellow ! BAIN, page : 166- No 7, 9, 21, 30 Time : 30’