EXAM P SAMPLE SOLUTIONS

74. Solution: E
First note R = 10/T . Then
⎡10 ⎤ ⎡ 10⎤ ⎛10
⎞
F R (r) = P[R ≤ r] = P
⎢
≤ r = P T ≥ = 1−FT⎜ ⎣T ⎥
⎦
⎢
⎣ r ⎥
⎟ . Differentiating with respect to
⎦ ⎝ r ⎠
⎛ ⎛10 ⎞⎞ ⎛ d ⎞⎛−10
⎞
r f R(r) = F′ R(r)
= d/dr ⎜1− FT ⎜ ⎟ =−⎜
FT () t ⎟⎜ 2
r
⎟
⎟⎠
⎝ ⎝ ⎠⎠ ⎝dt ⎠⎝ r
d 1
FT() t = fT() t = since T is uniformly distributed on [8, 12] .
dt
4
−1⎛−10⎞ 5
Therefore f R(r)
= ⎜ = 2 ⎟ . 2
4 ⎝ r ⎠ 2r
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75. Solution: A
Let X and Y be the monthly profits of Company I and Company II, respectively. We are
given that the pdf of X is f . Let us also take g to be the pdf of Y and take F and G to be
the distribution functions corresponding to f and g . Then G(y) = Pr[Y ≤ y] = P[2X ≤ y]
= P[X ≤ y/2] = F(y/2) and g(y) = G′(y) = d/dy F(y/2) = ½ F′(y/2) = ½ f(y/2) .
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76. Solution: A
First, observe that the distribution function of X is given by
x 3 1 x 1
F ( x) = ∫ dt =− | 4 3 1 = 1 − , x>
1
1
3
t t x
Next, let X1, X 2, and X3
denote the three claims made that have this distribution. Then if
Y denotes the largest of these three claims, it follows that the distribution function of Y is
given by
G y = Pr X ≤ y Pr X ≤ y Pr X ≤ y
( ) [ ] [ ] [ ]
1 2 3
⎛ 1 ⎞
= ⎜1 − , y 1
3 ⎟ >
⎝ y ⎠
while the density function of Y is given by
3
2 2
⎛ 1 ⎞ ⎛ 3 ⎞ ⎛ 9 ⎞⎛ 1 ⎞
g( y) = G'( y) = 3⎜1− 1 , y 1
3 ⎟ ⎜ 4 ⎟= ⎜ − 4 ⎟⎜ 3 ⎟ >
⎝ y ⎠ ⎝ y ⎠ ⎝ y ⎠⎝ y ⎠
Therefore,
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