EXAM P SAMPLE SOLUTIONS

111. Solution: E
3 f
Pr ⎡ ⎣1< Y < 3 X = 2⎤
⎦ = ∫ 1 f
y
dy
1
( − )
( 2, )
( 2)
2 ( ) 1
f ( 2, y) = y = y
4 2 1 2
∞
− 4−1 2−1 −3
∞
−3 −2
1 1 1
fx( 2)
= ∫ y dy =− y =
2 4 4
x
1
1
3
−3
y dy
1 2
−2
3 1 8
1 1 9 9
Finally, Pr ⎡ ⎣1< Y < 3 X = 2⎤ ⎦ = =− y = 1−
=
∫
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112. Solution: D
We are given that the joint pdf of X and Y is f(x,y) = 2(x+y), 0 < y < x < 1 .
x
∫
0
4
2
x
⎡
⎣
⎤
⎦0
Now fx(x) = (2x+ 2 y) dy = 2xy+
y = 2x 2 + x 2 = 3x 2 , 0 < x < 1
f ( xy , ) 2( x+ y) 2⎛1 y⎞
so f(y|x) = = = 2 ⎜ + 2 ⎟ , 0 < y < x
fx ( x) 3x 3⎝
x x ⎠
2⎡ 1 y ⎤ 2
f(y|x = 0.10) = + = [ 10 + 100y
3 ⎢
⎣0.1 0.01⎥ ] , 0 < y < 0.10
⎦ 3
0.05
2 ⎡20 100 2 ⎤ 0.05 1 1 5
P[Y < 0.05|X = 0.10] = ∫ [ 10 + 100ydy
] = y+ y = + =
3 ⎢
⎣ 3 3 ⎥
= 0.4167 .
⎦ 3 12 12
0
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113. Solution: E
Let
W = event that wife survives at least 10 years
H = event that husband survives at least 10 years
B = benefit paid
P = profit from selling policies
c
Then Pr[ H] = P[ H ∩ W] + Pr ⎡
⎣H ∩ W ⎤
⎦ = 0.96 + 0.01 = 0.97
and
Pr[ W ∩ H]
0.96
Pr[ W| H]
= = = 0.9897
Pr H 0.97
[ ]
c
Pr ⎡H ∩W
⎤
c
0.01
Pr ⎡W H
⎣ ⎦
⎣ | ⎤
⎦ = = = 0.0103
Pr 0.97
[ H ]
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