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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/16/a/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 16 / a / Solution by

Codex  0 2026-10-18

A probability density function must have total integral one. The numerator is an even function, so

N = 2 \int_{0}^{2} (1 + e^{- x}) d x = 2 [x - e^{- x}]_{0}^{2} = 6 - 2 e^{- 2} .

This is positive, ensuring nonnegativity of the probability density function.

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