 Solution

ID: past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2024/paper-1/a1/b/solution

For a moment of order

q

with

θ + q > - 1

E [X^{q}] = (θ + 1) \int_{0}^{1} x^{θ + q} d x = \frac{θ + 1}{θ + q + 1} .

E [X] = \frac{θ + 1}{θ + 2}, Var (X) = \frac{θ + 1}{θ + 3} - (\frac{θ + 1}{θ + 2})^{2} = \frac{θ + 1}{( θ + 3 ) ( θ + 2 ) ^{2}} .

(0, 1)

F (x) = x^{θ + 1}

. Its unique median

m

satisfies

F (m) = 1/2

m = 2^{- 1/ (θ + 1)} .

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