 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-resit/7/a/iii/solution

Write

r

for a positive integer moment index. The moment is

E [X^{r}] = θ c^{θ} \int_{c}^{\infty} x^{r - θ - 1} d x .

This improper integral converges exactly when

r - θ - 1 < - 1

, namely

r < θ

. For those indices,

E [X^{r}] = \frac{θ c ^{r}}{θ - r}, r \in Z_{> 0}, r < θ .

r = θ

the divergence is logarithmic; for

r > θ

it is a power divergence. Thus, in the source's moment notation, the positive integers

k

allowed are exactly

k < θ

. If

θ \leq 1

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