Year: 2009
Paper: 3
Question Number: 13
Course: UFM Statistics
Section: Central limit theorem
The vast majority of candidates (in excess of 95%) attempted at least five questions, and nearly a quarter attempted more than six questions, though very few doing so achieved high scores (about 2%). Most attempting more than six questions were submitting fragmentary answers, which, as the rubric informed candidates, earned little credit.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1488.4
Banger Comparisons: 1
\begin{questionparts}
\item The point $P$ lies on the circumference of a circle of unit radius and centre $O$. The angle, $\theta$, between $OP$ and the positive $x$-axis is a random variable, uniformly distributed on the interval $0\le\theta<2\pi$.
The cartesian coordinates of $P$ with respect to $O$ are $(X,Y)$.
Find the probability density function for $X$, and calculate $\var (X)$.
Show that $X$ and $Y$ are uncorrelated and discuss briefly whether they are independent.
\item The points $P_i$ ($i=1$, $2$, $\ldots$ , $n$) are chosen independently on the circumference of the circle, as in part (i), and have cartesian coordinates $(X_i, Y_i)$.
The point $\overline P$ has coordinates $(\overline X, \overline Y)$, where $\overline X =\dfrac1n \sum\limits _{i=1}^n X_i$ and $\overline Y =\dfrac1n \sum\limits _{i=1}^n Y_i$.
Show that $\overline X$ and $\overline Y$ are uncorrelated.
Show that, for large $n$, $\displaystyle \P\left(\vert \overline X \vert \le \sqrt{\frac2n}\right)\approx 0.95\,$.
\end{questionparts}\begin{questionparts}
\item $X = \cos \theta$ $\theta \sim U(0, 2\pi)$. Noting that $\mathbb{P}(X \geq t ) = \frac{2}{2\pi}\cos^{-1} t$ so $f_X(t) = \frac{1}{\pi} \frac{1}{\sqrt{1-x^2}}$
\begin{align*}
&& \E[X] &= 0 \tag{by symmetry} \\
&& \E[X^2] &= \int_0^{2\pi} \cos^2 \theta \frac{1}{2 \pi} \d \theta \\
&&&= \frac{1}{2} \cdot 2\pi \cdot \frac{1}{2\pi} \\
&&&= \frac12 \\
\Rightarrow & &\var[X] &= \frac12 \\
\\
&& \E[XY] &= \int_0^{2\pi} \cos \theta \sin \theta \frac{1}{2 \pi} \d \theta \\
&&&= \frac{1}{4\pi} \int_0^{2\pi} \sin 2\theta \d \theta \\
&&& =0 = \E[X]\E[Y]
\end{align*}
But note that clearly $X$ and $Y$ are not independent, since given $X$ there are only two possible values of $Y$.
\item $\,$ \begin{align*}
&& \E \left [ XY \right] &= \E \left [ \left ( \frac1n \sum_{i=1}^n X_i \right)\left ( \frac1n \sum_{i=1}^n Y_i\right) \right] \\
&&&= \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \E [X_i Y_j] \\
&&&= 0 = \E[X] \E[Y]
\end{align*}
Therefore $X$ and $Y$ are uncorrelated.
Note that $\E[X_i] = 0, \var[X_i] = \frac12$ so we can apply the central limit theorem to see that $X \approx N(0, \frac{1}{2n})$, in particular
\begin{align*}
&& 0.95 &\approx \mathbb{P}(|Z| < 2) \\
&&&= \mathbb{P} \left ( \Big |\frac{X}{\sqrt{\frac{1}{2n}}} \Big | < 2 \right ) \\
&&&= \mathbb{P}\left (|X| < \sqrt{\frac{2}{n}} \right)
\end{align*}
\end{questionparts}
A handful of candidates attempted this question with a couple making a good stab at part (i), but otherwise it was the odd crumb, if even that, which was collected.