EXAM P SAMPLE SOLUTIONS

5
( ) = ∑ ( ) Pr[
= ]
E⎡⎣f X ⎤⎦
f k X k
k = 1
⎛ 5 ⎞ ⎛ 4 ⎞ ⎛ 3 ⎞ ⎛ 2 ⎞ ⎛ 1 ⎞
= 100⎜ ⎟+ 200⎜ ⎟+ 300⎜ ⎟+ 325⎜ ⎟+ 350⎜
⎟
⎝15 ⎠ ⎝15 ⎠ ⎝15 ⎠ ⎝15 ⎠ ⎝15 ⎠
1 640
= [ 100 + 160 + 180 + 130 + 70] = = 213.33
3 3
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50. Solution: C
Let N be the number of major snowstorms per year, and let P be the amount paid to
n −3/2
(3/ 2) e
the company under the policy. Then Pr[N = n] =
, n = 0, 1, 2, . . . and
n!
⎧0
for N = 0
P = ⎨
.
⎩10,000(
N −1) for N ≥1
∞
n −3/2
(3/ 2) e
Now observe that E[P] = ∑10,000(
n −1)
n!
n=
1
∞
n −3/2
–3/2 (3/ 2) e
–3/2
= 10,000 e + ∑10,000(
n −1)
= 10,000 e + E[10,000 (N – 1)]
n=
0
n!
–3/2 –3/2
= 10,000 e + E[10,000N] – E[10,000] = 10,000 e + 10,000 (3/2) – 10,000 = 7,231 .
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51. Solution: C
Let Y denote the manufacturer’s retained annual losses.
⎧x
Then Y = ⎨
2
for 0.6 < x≤2
for x > 2
and E[Y] =
⎩
2 2.5 ∞
2.5 2
2.5 2.5
⎡2.5(0.6) ⎤ ⎡2.5(0.6) ⎤ 2.5(0.6) 2(0.6)
x⎢ dx 2
dx dx
x
⎥ + ⎢ = −
x
⎥
⎣ ⎦ ⎣ ⎦ x
x
∫ ∫ ∫
3.5 3.5 2.5 2.5 2
0.6 2 0.6
2.5 2.5 2.5 2.5 2.5
2.5(0.6) 2 2(0.6) 2.5(0.6) 2.5(0.6) (0.6)
= − + =− + + = 0.9343 .
1.5 0.6
2.5 1.5 1.5 1.5
1.5 x (2) 1.5(2) 1.5(0.6) 2
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∞