 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/16/a/vi/solution

E [S^{2}] = \int_{0}^{r} \frac{s ^{2}}{R} d s = \frac{r ^{3}}{3 R} .

Subtracting the squared expected value gives the variance and standard deviation:

Var (S) = \frac{r ^{3}}{3 R} - \frac{r ^{4}}{4 R ^{2}} = \frac{r ^{3} ( 4 R - 3 r )}{12 R ^{2}}, σ = \frac{r ^{3} ( 4 R - 3 r )}{12 R ^{2}} .

The variance is nonnegative because

0 < r \leq R

. Both

μ

and

σ

r = R

the result reduces to

R / 12

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