 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/14/a/solution

Write

P_{k} = P (X = k)

for

k = 0, 1, 2, \dots

, with

\sum_{k} P_{k} = 1

. Provided the relevant moments exist, the expected value and variance are

μ = k = 0 \sum \infty k P_{k}, σ^{2} = k = 0 \sum \infty (k - μ)^{2} P_{k} = k = 0 \sum \infty k^{2} P_{k} - μ^{2} .

The displayed support includes zero, so these are nonnegative integer values.

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