Year: 2023
Paper: 2
Question Number: 12
Course: LFM Stats And Pure
Section: Continuous Probability Distributions and Random Variables
No solution available for this problem.
Many candidates were able to express their reasoning clearly and presented good solutions to the questions that they attempted. There were excellent solutions seen for all of the questions. An area where candidates struggled in several questions was in the direction of the logic that was required in a solution. Some candidates failed to appreciate that separate arguments may be needed for the "if" and "only if" parts of a question and, in some cases, candidates produced correct arguments, but for the wrong direction. In several questions it was clear that candidates who used sketches or diagrams generally performed much better that those who did not. Sketches often also helped to make the solution clearer and easier to understand. Several questions on the STEP papers ask candidates to show a given result. Candidates should be aware that there is a need to present sufficient detail in their solutions so that it is clear that the reasoning is well understood.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Each of the independent random variables $X_1, X_2, \ldots, X_n$ has the probability density function $\mathrm{f}(x) = \frac{1}{2}\sin x$ for $0 \leqslant x \leqslant \pi$ (and zero otherwise). Let $Y$ be the random variable whose value is the maximum of the values of $X_1, X_2, \ldots, X_n$.
\begin{questionparts}
\item Explain why $\mathrm{P}(Y \leqslant t) = \big[\mathrm{P}(X_1 \leqslant t)\big]^n$ and hence, or otherwise, find the probability density function of $Y$.
\end{questionparts}
Let $m(n)$ be the median of $Y$ and $\mu(n)$ be the mean of $Y$.
\begin{questionparts}
\setcounter{enumi}{1}
\item Find an expression for $m(n)$ in terms of $n$. How does $m(n)$ change as $n$ increases?
\item Show that
\[\mu(n) = \pi - \frac{1}{2^n}\int_0^{\pi} (1-\cos x)^n\,\mathrm{d}x\,.\]
\begin{enumerate}
\item[(a)] Show that $\mu(n)$ increases with $n$.
\item[(b)] Show that $\mu(2) < m(2)$.
\end{enumerate}
\end{questionparts}
This was the more popular of the two "Probability and Statistics" questions and a larger number of substantial attempts was seen. Part (i) was generally completed well although in some cases there was insufficient explanation that "Y ≤ t" is equivalent to "Xi ≤ t for all i". Many candidates successfully calculated the value of m(n) for part (ii), but some only stated that m(n) increases, rather than considering the value of the limit. In part (iii) many candidates successfully showed the formula for μ(n). A number of candidates attempted to prove that μ(n) is increasing by differentiating with respect to n and showing that this is a positive quantity. However, none of these candidates were able to produce a fully correct version of this approach. In part (iii)(b) most candidates were able to calculate μ(2) correctly, but then a number of errors were seen in the subsequent argument. Common errors were to fail to consider which choice of square root is appropriate and to omit to consider the effect of squaring on an inequality.