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

The probability density is symmetric about $300\mathrm{mL}$, vanishes at $270$ and $330\mathrm{mL}$, and has its single maximum $\pi /(120\mathrm{mL})$ at $300\mathrm{mL}$. It is zero outside that interval. Its area is one: with the dimensionless change of variablesnormalization is $\frac{1}{2}{\int }_{-\pi /2}^{\pi /2}\cos xdx=1$.

Passing corresponds to $-\pi /6\le X\le \pi /6$ in the normalized variable from part (i). Integrating its probability density,The complementary probability of detecting the fault in one test is also $1/2$.

The absolutely convergent geometric series has sum ${\sum }_{k=0}^{\infty }{r}^k=(1-r{)}^{-1}$ for $0<r<1$. Its Cauchy product with itself isThere are exactly $m+1$ pairs of nonnegative integers whose sum is $m$. Relabeling $j=m+1$ provesAbsolute convergence justifies the multiplication and regrouping; no differentiation of an infinite series is needed.

Let $T$ count tests up to and including the first failure. By independence of random variables, the event $T=j$ requires $j-1$ passes and then a failure. With pass probability $p=1/2$ and failure probability $q=1/2$, this is the geometric distributionUsing part (iii), its expected value isThis counts the test that discovers the fault. The expected number of earlier passing tests alone would be one.