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MTH3402 Exercise 5
发布时间:2022-02-07
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MTH3402
Exercise 5
1. The joint pdf ofX and Y is f x, y
; x
0, y
0 . Find E
X2
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
e
x1
x2
; x1 , x2
0 . Let Y1
X1
X2 and
Y2
1 , find the joint pdf of Y1 and Y2 .
.
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
X
Y and Z
X .