 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/c/ii/solution

Use the law of total probability to average the conditional interval probability over

Y

P (z \leq Z \leq z + d z) = \int_{- \infty}^{\infty} g (y) \int_{z}^{z + d z} f (v - y) d v d y .

The integrands are nonnegative, so Tonelli's theorem permits interchanging the integrals. Thus

P (z \leq Z \leq z + d z) = \int_{z}^{z + d z} [\int_{- \infty}^{\infty} g (y) f (v - y) d y] d v .

h (z) = \int_{- \infty}^{\infty} g (y) f (z - y) d y .

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