 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/14/a/i/solution

The probability density is symmetric about

300 mL

, vanishes at

270

and

330 mL

, and has its single maximum

π / (120 mL)

300 mL

. It is zero outside that interval. Its area is one: with the dimensionless change of variables

X = \frac{π ( v - 300 mL )}{60 mL}, p_{X} (x) = \frac{1}{2} cos x, - \frac{π}{2} \leq x \leq \frac{π}{2},

normalization is

\frac{1}{2} \int_{- π /2}^{π /2} cos x d x = 1

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