 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/discrete-mathematics-continuous-mathematics-and-probability/8/b/iii/solution

Choose $g (t) = e^{t}$ . If

W

r

, then for

c < r

E [e^{c W}] = \int_{0}^{\infty} r e^{- (r - c) t} d t = \frac{r}{r - c},

while the integral diverges to

+ \infty

when

c \geq r

. Since

Z

has rate

3

E [g (Z)] = \frac{3}{2}, E [g (Z)^{2}] = 3, Var (g (Z)) = \frac{3}{4} .

For the other two rates,

E [g (V)] = 2/ (2 - 1) = 2

, whereas

E [g (U)] = \int_{0}^{\infty} 1 d t = + \infty

. These computations establish exactly the required finite variance, finite expected value, and infinite expected value.

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