STAT200C Assignment 3

Author

Zhaoxia Yu

Important

This material is for the use of the students enrolled in the course. It is protected by United States copyright law [Title 17, U.S. Code]. It should not be distributed without written permission from the instructor Zhaoxia Yu. In particular, please do not upload it to online “resource” websites.

  1. Let \(X\sim N_3(0,\Sigma)\) with

\[ \Sigma= \begin{pmatrix} 1 & \rho & \rho \\ \rho & 1 & \rho \\ \rho & \rho & 1 \end{pmatrix} \]

and \(\rho > -1/2\).

  1. Give the multiple correlation between the first variable and the second two. For this specific problem, when is it one? When is it zero?

  2. Give the joint distribution of the first two variables given the third.

  3. What is the correlation of the first two variables given the third? What is the largest possible conditional correlation? Justify.

  4. Show that \[ \sum_{i=1}^3 (X_i-\bar X)^2/(1-\rho) \] follows a chi-square distribution and give the degrees of freedom. Feel free to rescale itS if necessary.

  1. Let \(\mathbf{X} = (X_1,\cdots,X_n)' \sim N_n(\mu\mathbf{1},\sigma^2\mathbf{I})\), where \(\mathbf{1} = \mathbf{1}_n = (1,\cdots,1)'\) is an \(n\)-vector of ones and \(\mathbf{I}\) is an \(n \times n\) identity matrix. Let

\[ Q_1 = \sum_{i=1}^n (X_i - \bar{X})^2 \]

and

\[ Q_2 = \frac{1}{n}\left(\sum_{i=1}^n X_i \right)^2 \]

where, as usual,

\[ \bar{X} = \sum_{i=1}^n X_i /n. \]

  1. Write \(Q_1\) and \(Q_2\) as quadratic forms \(\mathbf{X}'\mathbf{A}_1\mathbf{X}\) and \(\mathbf{X}'\mathbf{A}_2\mathbf{X}\), respectively. (Hint: note that we can write \(\sum_{i=1}^n X_i = \mathbf{1}'\mathbf{X}\)).

  2. Show that \(\mathbf{A}_1\) and \(\mathbf{A}_2\) are projection matrices.

  3. What are the ranks of \(\mathbf{A}_1\) and \(\mathbf{A}_2\)?

  4. What is the distribution of \(Q_1/\sigma^2\)? If necessary, you can re-scale it.

  5. What is the distribution of \(Q_2/\sigma^2\)? If necessary, you can re-scale it.

  6. Are \(Q_1\) and \(Q_2\) independent? Justify your answer.

  1. Let \(\mathbf{X}\sim N_k(\boldsymbol{\mu},\mathbf{D})\) with \(\mathbf{D}=diag(\sigma_1^2,\cdots,\sigma_k^2)\) and \(rank(\mathbf{D})=k\). What is the noncentrality parameter \(\lambda\) (as a function of \(\mu\)’s and \(\sigma\)’s) for the distribution of \(\mathbf{X'D^{-1}X}\)?

  2. Suppose that \(\mathbf D\) is a positive definite \(k\times k\) matrix. Let

\[ \mathbf{A}=\mathbf{D}^{-1}-(\mathbf{D}^{-1}\mathbf{11'D}^{-1})/\mathbf{1'D}^{-1}\mathbf{1}. \]

If \(\mathbf{X}\sim N_k(0,\mathbf{D})\), what is the distribution of \(\mathbf{X'AX}\)?

  1. Suppose that \(K\sim Poisson(\lambda/2)\) and \(X|K \sim \chi_{p+2k}^2\). We mentioned in class that \(X\sim \chi_p^2 (\lambda)\). Prove this.

Hint: Let \(Z_i\)’s be independent and \(Z_i\sim N(\mu_i,1)\). Compute the mgf of \(Z_i^2\) then the mgf of \(\sum Z_i^2\). By doing this, you obtain the mgf for a non-central chi-square distribution with \(p\) df and \(\lambda=\sum \mu_i^2\). Next compute the mgf of \(X\) and compare it to that of \(\chi_p^2(\lambda)\).