MTH3402 Exercise 5
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MTH3402
Exercise 5
1. The joint pdf ofX and Y is f x, y ; x 0, y 0 . Find EX2 Y y .
2. pdf ofX and Y is f x, y ; y 0, 0 x y .
3. Let X and Y are independent random variables from binomial distributions with parameters
n1 , p and n2 , p , respectively. Find the joint mgf of W X Y . What is the distribution of W?
4. Find the joint mgf of Y X1 X2 ... Xn if X1 , X2 ,..., Xn are independent Poisson random variables with mean 1 , 2 ,..., n . What is the distribution of Y?
5. Find the joint mgf of Z X1 X2 ... Xn if X1 , X2 ,..., Xn are independent exponential
random variables with parameter . Show that Z has gamma distribution with mean .
6. If the pdf ofX is p x 0.5x , x 1, 2,3,... . find the pdf of Y X3 .
7. The pdf ofX is f x 6x 1 x , 0 x 1 . Find the pdf of Y X3 .
1 X
9. Let the joint pdf of X1 and X2 be f x1 , x2 ex1x2 ; x1 , x2 0 . Let Y1 X1 X2 and
Y2 1 , find the joint pdf of Y1 and Y2 .
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10. Let the joint pdf of X and Y be f x, y 24xy ; 0 x 1, 0 y 1, x y 1 . Find the joint pdf of W and Z using the transformation W XY and Z X .
2022-02-07