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test midterm answers - EE528 Probability and Random Process...

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EE528 Probability and Random Process Midterm Test March 23, 2010 Problem 1 (20 pt). and are discrete random variables with their joint pdf (or pmf): ( ) 1,2 1,2 ( , ) 0 i) Find the value of . ii) Find the correlation coefficient between XY X Y k x y for x and y p x y otherwise k + = = = and . (Note. Students must show the procedure of finding the answer.) X Y Problem 2 (20 pt). Find the pdf and the moment generating function of the following random variable . i) is an exponential random variable with parameter (the arrival rate). ii) is an -erlang ra X X X m λ ndom variable with parameter . iii) is a gamma random variable with parameter and . iv) is a chi-square random variable with degrees of freedom. X X k λ λ α ( ) 1 i) is an exponential random variable with parameter (the arrival rate). ( ) for 0. ( ) for ii) is an -erlang random variable with parameter . ( ) for ( 1)! x X X m x X X f x e x M t t t X m x e f x x m λ λ λ λ λ λ λ λ λ λ = = < = ( ) 1 0. ( ) for iii) is a gamma random variable with parameter and .
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