Problems

Filters
Clear Filters

2 problems found

2010 Paper 3 Q13
D: 1700.0 B: 1516.0
correlation covariance bivariate data linear combinations of random variables independent random variables expectation variance Cholesky decomposition

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 \(\mathrm{Corr}(X_i,X_j) = \rho_{ij}\).

Solution: \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 \(X_i = \mu_i + \sigma_i Y_i\)

View
2009 Paper 3 Q13
D: 1700.0 B: 1488.4
central limit theorem probability density function uniform distribution variance correlation independence trigonometric derived distribution arcsine distribution normal approximation

  1. 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.
  2. 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 \(n\), \(\displaystyle \P\left(\vert \overline X \vert \le \sqrt{\frac2n}\right)\approx 0.95\,\).

Solution:

  1. \(X = \cos \theta\) \(\theta \sim U(0, 2\pi)\). 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 \(Y\).
  2. \(\,\) \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*}

View