Year: 2010
Paper: 3
Question Number: 13
Course: UFM Statistics
Section: Bivariate data
About 80% of candidates attempted at least five questions, and well less than 20% made genuine attempts at more than six. Those attempting more than six questions fell into three camps which were those weak candidates who made very little progress on any question, those with four or five fair solutions casting about for a sixth, and those strong candidates that either attempted 7th or even 8th questions as an "insurance policy" against a solution that seemed strong but wasn't, or else for entertainment!
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1516.0
Banger Comparisons: 1
In this question, ${\rm Corr}(U,V)$ denotes the product moment correlation coefficient between the random variables $U$ and $V$, defined by
\[
\mathrm{Corr}(U,V) \equiv \frac{\mathrm{Cov}(U,V)}{\sqrt{\var(U)\var(V)}}\,.
\]
The independent random variables $Z_1$, $Z_2$ and $Z_3$ each have expectation 0 and variance 1. What is the value of $\mathrm{Corr} (Z_1,Z_2)$?
Let $Y_1 = Z_1$ and let
\[
Y_2 = \rho _{12} Z_1 +
(1 - {\rho_{12}^2})^{ \frac12} Z_ 2\,,
\]
where $\rho_{12}$ is a given constant with $-1<\rho
_{12}<1$.
Find $\E(Y_2)$, $\var(Y_2)$ and $\mathrm{Corr}(Y_1, Y_2)$. Now let $Y_3 = aZ_1 + bZ_2 + cZ_3$, where $a$, $b$ and $c$ are real constants and $c\ge0$. Given that $\E(Y_3) = 0$, $\var(Y_3) = 1$,
$ \mathrm{Corr}(Y_1, Y_3) =\rho^{{2}}_{13} $ and $ \mathrm{Corr}(Y_2, Y_3)= \rho^{{2}} _{23}$, express $a$, $b$ and $c$ in terms of $\rho^{2} _{23}$, $\rho^{2}_{13}$ and $\rho^{2} _{12}$. Given constants $\mu_i$ and $\sigma_i$, for $i=1$, $2$ and $3$, give expressions in terms of the $Y_i$ for random variables $X_i$ such that $\E(X_i) = \mu_i$, $\var(X_i) = \sigma_ i^2$ and $\mathrm{Corr}(X_i,X_j) = \rho_{ij}$.\begin{align*}
\mathrm{Corr} (Z_1,Z_2) &= \frac{\mathrm{Cov}(Z_1,Z_2)}{\sqrt{\var(Z_1)\var(Z_2)}} \\
&= \frac{\mathbb{E}(Z_1 Z_2)}{\sqrt{1 \cdot 1}} \\
&= \frac{\mathbb{E}(Z_1)\mathbb{E}(Z_2)}{\sqrt{1 \cdot 1}} \\
&= \frac{0}{1} \\
&= 0
\end{align*}
\begin{align*}
&& \mathbb{E}(Y_2) &= \mathbb{E}(\rho_{12} Z_1 +
(1 - {\rho_{12}^2})^{ \frac12} Z_ 2) \\
&&&= \mathbb{E}(\rho_{12} Z_1) + \mathbb{E}(
(1 - {\rho_{12}^2})^{ \frac12} Z_ 2) \\
&&&= \rho_{12}\mathbb{E}( Z_1) + (1 - {\rho_{12}^2})^{ \frac12}\mathbb{E}(
Z_ 2) \\
&&&= 0\\
\\
&& \textrm{Var}(Y_2) &= \textrm{Var}(\rho _{12} Z_1 +
(1 - {\rho_{12}^2})^{ \frac12} Z_ 2) \\
&&&= \textrm{Var}(\rho_{12} Z_1)+\textrm{Cov}(\rho_{12} Z_1,(1 - {\rho_{12}^2})^{ \frac12} Z_ 2 ) + \textrm{Var}((1 - {\rho_{12}^2})^{ \frac12} Z_ 2) \\
&&&= \rho_{12}^2\textrm{Var}( Z_1)+\rho_{12} (1 - {\rho_{12}^2})^{ \frac12} \textrm{Cov}(Z_1, Z_ 2 ) + (1 - {\rho_{12}^2})\textrm{Var}(Z_ 2) \\
&&&= \rho_{12}^2 + (1-\rho_{12}^2) = 1 \\
\\
&& \textrm{Cov}(Y_1, Y_2) &= \mathbb{E}((Y_1-0)(Y_2-0)) \\
&&&= \mathbb{E}(Z_1 \cdot (\rho _{12} Z_1 +
(1 - {\rho_{12}^2})^{ \frac12} Z_ 2)) \\
&&&= \rho_{12} \mathbb{E}(Z_1^2) + (1-\rho_{12}^2)^{\frac12}\mathbb{E}(Z_1, Z_2) \\
&&&= \rho_{12} \\
\Rightarrow && \textrm{Corr}(Y_1, Y_2) &= \frac{\textrm{Cov}(Y_1, Y_2)}{\sqrt{\textrm{Var}(Y_1)\textrm{Var}(Y_2)}} \\
&&&= \frac{\rho_{12}}{1 \cdot 1} = \rho_{12}
\end{align*}
Suppose $Y_3 =aZ_1 +bZ_2+cZ_3$ with $\mathbb{E}(Y_3) = 0$ (must be true), $\textrm{Var}(Y_3) = 1 = a^2+b^2+c^2$ and $\textrm{Corr}(Y_1, Y_3) = \rho_{13}, \textrm{Corr}(Y_2, Y_3) = \rho_{23}$.
\begin{align*}
&& \textrm{Corr}(Y_1,Y_3) &= \textrm{Cov}(Y_1, Y_3) \\
&&&= \textrm{Cov}(Z_1, aZ_1 +bZ_2+cZ_3) \\
&&&= a \\
\Rightarrow && a &= \rho_{13} \\
\\
&& \textrm{Corr}(Y_2,Y_3) &= \textrm{Cov}(Y_2, Y_3) \\
&&&= \textrm{Cov}(\rho_{12}Z_1+(1-\rho_{12}^2)^\frac12Z_2, \rho_{13}Z_1 +bZ_2+cZ_3) \\
&&&= \rho_{12}\rho_{13}+(1-\rho_{12}^2)^\frac12b \\
\Rightarrow && \rho_{23} &= \rho_{12}\rho_{13}+(1-\rho_{12}^2)^\frac12b \\
\Rightarrow && b &= \frac{\rho_{23}-\rho_{12}\rho_{13}}{(1-\rho_{12}^2)^\frac12} \\
&& c &= \sqrt{1-\rho_{13}^2-\frac{(\rho_{23}-\rho_{12}\rho_{13})^2}{(1-\rho_{12}^2)}}
\end{align*}
Finally, let $X_i = \mu_i + \sigma_i Y_i$
This was the least popular question with little more than a couple of handfuls of attempts. In view of the small number of attempts, there were no detectable trends though oddly, the very few candidates who mastered this question conspired to avoid full marks by making minor algebraic inaccuracies having dealt with all the trickier aspects.