 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability-resit/9/b/i/solution

x

and integrate over its support:

E [X] = \frac{Γ ( α + β )}{Γ ( α ) Γ ( β )} \int_{0}^{1} x^{α} (1 - x)^{β - 1} d x = \frac{Γ ( α + β )}{Γ ( α ) Γ ( β )} \frac{Γ ( α + 1 ) Γ ( β )}{Γ ( α + β + 1 )} = \frac{α}{α + β} .

The middle equality is the beta function integral with parameters

α + 1

and

β

; the last uses the recurrence for the gamma function. Both parameters are positive, so the integral converges. Equivalently, multiplying the beta distribution probability density function by

x

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