 Solution

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/ii/solution

Reversing sign reverses the inequality. Since

X

F_{Y} (y) = P (X \geq - y) = 1 - P (X < - y) = 1 - F_{X} (- y) .

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

F_{X}

cancels the minus sign from the chain rule. If point masses were allowed, the cumulative distribution function would instead require the left limit

1 - F_{X} ((- y)^{-})

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