 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/ii/solution

For

q > 1

, decreasing the second parameter by one gives another valid beta distribution density and

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

Normalization therefore gives

E [\frac{1}{1 - X}] = \frac{p + q - 1}{q - 1} .

The restriction

q > 1

is essential for this finite expected value; when

q = 1

, the singularity near

1

instead makes the expectation infinite.

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