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

Table of contents
- Solution i

Solution

Multiply the probability density function by

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

converts its kernel to that of a beta distribution probability density function with its first parameter increased by one.^[ref]

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