 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability/7/a/i/solution

x

gives a constant multiple of the normalized density with parameters

(p + 1, q)

x f_{p, q} (x) = \frac{p}{p + q} f_{p + 1, q} (x) .

Integrating and using the fact that the latter probability density function integrates to one gives

E [X] = \frac{p}{p + q} .

This is a normalization-ratio argument, requiring no explicit integration of the polynomial density.

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