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

Table of contents
- Solution i

Solution

The probability density is symmetric about

300 mL

, vanishes at

270

and

330 mL

, and has its single maximum

π / (120 mL)

at

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

.

Figure 1.
Coffee-volume density and the two-cup tail region
. Original graphs. Left: the density with the passing interval shaded. Right: transformed independent cup volumes $X, Y$ and the triangular region $X + Y \geq π /2$ used for the probability in part (b). Details.

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