Assignment 2

[This assignment depends on Distribution Theory Chapters 3 and 4.]

1. Independent variables $X$ and $Y$ have pdf's as follows
   
   $$f_{X}(x)=12x^2(1-x), \ 0 \leq x \leq 1,$$
   
   
   
   
   $$f_{Y}(y)=2y, \ \ 0 \leq y \leq 1.$$
   
   
   Find the distribution of the random variable $U$ defined by $U=XY$.
2. Random variables $X$ and $Y$ are jointly distributed as follows:
   
   $$f_{X,Y}(x,y)=24y(1-x), \ 0 \leq y \leq x \leq 1.$$
   
   
   Find the distribution of Z defined by $Z=Y/X$.
3. Random variables $X$ and $Y$ are independent with pdf's
   
   $$f_{X}(x)=e^{-x}, \ x \geq 0,$$
   
   
   
   
   $$f_{Y}(y)=y e^{-y}, \ y \geq 0.$$
   
   
   Find the distributions of $X/Y$ and $X+Y$.
4. Given $2$ random observations of the variable $X$ with pdf
   
   $$f_{X}(x)=1/x^2, \ 1 \leq x < \infty,$$
   
   
   find the joint and marginal distributions of $U$ and $V$ defined by
   
   $$U=X_1X_2 \mbox{ and }V=X_1/X_2.$$
   
   

5. If random variables $X$ and $Y$ have joint pdf given by
   
   $$f_{X,Y}(x,y)=3x, \ 0 \leq y \leq x \leq 1,$$
   
   
   find the pdf of $Z$ where $Z=X-Y$.
6. 1. Find $\Sigma$ where $\Sigma$$={\bf A}^{-1}$ and
      
      $$A= \left[\begin{array}{rrr} 7 & 3 & 2\\ 3 & 4 & 1\\ 2 & 1 & 2 \end{array} \right]$$
      
      

   2. If $\Sigma$ is the covariance matrix for the random variables $X_1$, $X_2$, $X_3$ which have means $1$, $1$, $2$ respectively, write down the multivariate normal density function, $f_{{\bf X}}{\bf (x)}$.
   3. Write down the marginal pdf's $f_{X_1}(x_1)$, $f_{X_3}(x_3)$.
   4. If random variable $Y$ is defined by
      
      $$Y=X_1-2X_2-X_3,$$
      
      
      find $E(Y)$ and Var(Y).
7. Random variables $X_1$, $X_2$, $X_3$ have a multivariate normal distribution with quadratic form $Q_1$ in the pdf, where
   
   $$Q_1=2x_1^2+x_2^2+4x_3^2-x_1x_2-2x_1x_3.$$
   
   
   1. Find the variances of $X_1$, $X_2$, $X_3$.
   2. Find cov($X_1,\,X_2$), cov($X_1, \,X_3$), cov($X_2, \, X_3$).
   3. What is the marginal pdf of $X_2$?
8. If $X_1$, $X_2$, $X_3$ are each distributed as N(0,1) and are mutually independent, show that the quadratic forms
   $$\begin{eqnarray*} Q_1&=&4X_1^2+4X_2^2+X_3^2-8X_1X_2-4X_2X_3+4X_1X_3\\ Q_2&=&11X_1^2+14X_2^2+20X_3^2+20X_1X_2+16X_2X_3-4X_1X_3 \end{eqnarray*}$$
   
   
   
   
   
   are independent and find their separate expected values.
9. If X $\sim$ $N_p$($\mu$, $\Sigma$), show that Y$=$ CX is distributed N(C$\mu$, C$\Sigma$C$'$) where C is a $p \times p$ non-singular matrix.
10. Suppose X is a random vector (not necessarily normally distributed) with $p$ components with
    
    $$\mbox{E({\bf X})}={\bf0} \mbox{ and cov}({\bf X})= \sigma^2{\bf I},$$
    
    
    show that
    
    $$E({\bf X}'{\bf A}{\bf X})=\sigma^2 \mbox{tr}({\bf A}).$$
    
    
    [Does your answer to 8. agree with this?]
11. $X_1$, ..., $X_p$ are normal random variables which have zero means and covariance matrix $\Sigma$. Show that a necessary and sufficient condition for the independence of the quadratic forms X$'$BX and X$'$CX is B$\Sigma$C$=$0.