 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/4/b/i/solution

Since

Y \geq 1

, its finite mean satisfies

1 \leq μ < \infty

. Every

q_{k} = k p_{k} / μ

is nonnegative, and

\sum_{k \geq 1} q_{k} = \frac{1}{μ} \sum_{k \geq 1} k p_{k} = \frac{E Y}{μ} = 1.

Thus

(q_{k})

is a probability mass function. It defines the size-biased distribution of

Y

: an outcome of size

k

receives

k

times its original relative weight.

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