Year: 2022
Paper: 3
Question Number: 12
Course: LFM Stats And Pure
Section: Continuous Probability Distributions and Random Variables
No solution available for this problem.
One question was attempted by well over 90% of the candidates two others by about 90%, and a fourth by over 80%. Two questions were attempted by about half the candidates and a further three questions by about a third of the candidates. Even the other three received attempts from a sixth of the candidates or more, meaning that even the least popular questions were markedly more popular than their counterparts in previous years. Nearly 90% of candidates attempted no more than 7 questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item The point $A$ lies on the circumference of a circle of radius $a$ and centre $O$. The point $B$ is chosen at random on the circumference, so that the angle $AOB$ has a uniform distribution on $[0, 2\pi]$. Find the expected length of the chord $AB$.
\item The point $C$ is chosen at random in the interior of a circle of radius $a$ and centre $O$, so that the probability that it lies in any given region is proportional to the area of the region. The random variable $R$ is defined as the distance between $C$ and $O$.
Find the probability density function of $R$.
Obtain a formula in terms of $a$, $R$ and $t$ for the length of a chord through $C$ that makes an acute angle of $t$ with $OC$.
Show that as $C$ varies (with $t$ fixed), the expected length $\mathrm{L}(t)$ of such chords is given by
\[ \mathrm{L}(t) = \frac{4a(1-\cos^3 t)}{3\sin^2 t}\,. \]
Show further that
\[ \mathrm{L}(t) = \frac{4a}{3}\left(\cos t + \tfrac{1}{2}\sec^2(\tfrac{1}{2}t)\right). \]
\item The random variable $T$ is uniformly distributed on $[0, \frac{1}{2}\pi]$. Find the expected value of $\mathrm{L}(T)$.
\end{questionparts}
The least popular question on the paper, it was also the least successful with a mean score of just under one quarter marks. Many attempts did not make much progress beyond the first part. Candidates with a good understanding of how to calculate the expectation of a function of a random variable generally made very good progress. In part (i) many candidates were able to calculate the length of the chord, although many used the cosine rule on an isosceles triangle to reach a√(2−2cos2θ), making that integration a little harder. A significant number of candidates who attempted this part omitted to include the probability density function when integrating to calculate the expected value. A small number of candidates chose to consider the length of the chord as a random variable and calculated its probability density function, from which they could then calculate the expected value. While this approach was in general successful it was a significantly more complicated approach. In part (ii), many candidates were able to work out the probability density function. Several candidates struggled to find an expression for the length of the chord and so failed to make any further progress from this point. Those that did were often able to complete the calculation of the expected value correctly. In a small number of cases, candidates attempted to calculate the probability density function for the length of the chord in order to calculate the expected value. In this case care needs to be taken with the limits of the integration as the shortest possible length for such chords needs to be calculated. A good number of candidates were able to rearrange the expected value in part (ii) into the requested form and many were then able to complete part (iii) successfully, although a number of attempts again omitted the probability density function and other attempts multiplied the function by t before integrating.