EXAM P SAMPLE SOLUTIONS

120. Solution: A
We are given that X denotes loss. In addition, denote the time required to process a claim
by T.
⎧3
2 1 3
⎪ x ⋅ = x, x< t < 2 x,0≤ x≤
2
Then the joint pdf of X and T is f( x, t) = ⎨8 x 8
⎪⎩ 0, otherwise.
Now we can find P[T ≥ 3] =
4 2 4 2
3 ⎡ 3 2⎤ ∫∫ xdxdt =
8 ∫⎢ x
16 3 t /2 3⎣ ⎥
⎦t/2 = 11/64 = 0.17 .
4
4
⎛12 3 2⎞ ⎡1213⎤ dt = ∫⎜
− t ⎟dt= − t
16 64 16 64 3⎝
⎠ ⎣
⎢
⎦
⎥
3
12 ⎛36
27 ⎞
= −1−⎜ − ⎟
4 ⎝16
64 ⎠
t t=2x
4
3
2
1
1 2
t=x
x
--------------------------------------------------------------------------------------------------------
121. Solution: C
The marginal density of X is given by
1
3
1 1 1
1
2 ⎛ xy ⎞ ⎛ x⎞
fx( x) = ∫ ( 10 − xy ) dy = ⎜10y− ⎟ = 10
0
⎜ − ⎟
64 64 ⎝ 3 ⎠ 64 ⎝ 3 ⎠
Then
2
10 10 1 ⎛ x ⎞
∫2 x ∫ ⎜ ⎟ x
2
E( X) = xf ( x) dx= 10x−
d
64 ⎝ 3 ⎠
= 1 1000 8
⎡⎛ ⎞ ⎛ ⎞⎤
500 20
64
⎢⎜ − ⎟−⎜ − ⎟
9 9
⎥ = 5.778
⎣⎝ ⎠ ⎝ ⎠ ⎦
0
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10
3
⎛ 2 x ⎞
1
= ⎜5x−⎟ 64 ⎝ 9 ⎠
2