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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability-resit/9/b/i/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 9 / b / i / Solution by

Codex  0 2026-10-31

Translate the inequality inside the cumulative distribution function:

F_{Y} (y) = P (1 + X \leq y) = P (X \leq y - 1) = F_{X} (y - 1) .

Differentiating with the chain rule gives the probability density function

f_{Y} (y) = f_{X} (y - 1)

almost everywhere. Translation shifts the probability distribution without changing its shape or its total probability.

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