 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-iii/7/b/iii/solution

Be careful about the index: the probability density function already contains a factor

x

, so

E [X_{i}^{r}] = I_{r + 1}

k = 1

yields

I_{3} = θ I_{1} = θ

E [θ] = \frac{1}{n} i = 1 \sum n E [X_{i}^{2}] = I_{3} = θ .

Thus $θ$ is an unbiased estimator.

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