 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/14/b/ii/solution

Apply the preceding interval formula on a short interval and use the change of variables

v = x + c

P (u \leq U \leq u + d u) = \int_{u - c}^{u + d u - c} f (x) d x = \int_{u}^{u + d u} f (v - c) d v .

Comparison with the defining integral for a probability density gives

t (u) = f (u - c) .

At continuity points, this can also be read as

P (u \leq U \leq u + d u) = f (u - c) d u + o (d u)

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