 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/18/a/solution

Write

p_{j} = P (X = j)

, with

\sum_{j = 0}^{N} p_{j} = 1

μ = j = 0 \sum N j p_{j}, σ^{2} = j = 0 \sum N (j - μ)^{2} p_{j} = j = 0 \sum N j^{2} p_{j} - μ^{2}, P (X < Y) = j = 0 \sum Y - 1 p_{j} .

The upper summation limit

Y - 1

is essential because the inequality is strict and the variable is integer-valued.

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