Assignment 1

[This assignment covers the work in Distribution Theory Chapters 1 and 2.]

Do matrix calculations in R using the functions solve() for the inverse and eigen() for the eigen values. The determinant of A is calculated by the product of eigen values - prod(eigen(A)$values). Use the help files for the details.

1. If $X_1, X_2, X_3$ are identically and independently distributed (iid) random variables with probability density function (pdf) $f(x)$, $x \in [0, \infty)$, and cumulative distribution function (cdf) $F(x)$, find
   
   $$P(X_1 < X_2 < X_3 \ \mbox{ or }\ X_1 > X_2 > X_3).$$
   
   

2. Assuming that the conditional pdf of $Y$ given X=x is
   
   $$f_{Y\vert X=x}(y)=\left \{ \begin{array}{ll} 2y/x^2 & \mbox{if \ $0<y<x<1$}\\ 0 & \mbox{otherwise} \end{array} \right.$$
   
   
   and the pdf of $X$ is
   
   $$f_X(x)=4x^3, \ \ 0 < x < 1,$$
   
   
   find
   1. the joint pdf of $X$ and $Y$;
   2. the marginal pdf of $Y$;
   3. $P(X>2Y)$.
3. Random variables $X$ and $Y$ have joint pdf
   
   $$f_{X,Y}(x,y) = \left\{ \begin{array}{ll} 1 & \mbox{ if $-y < ... ...d \ $ 0 < y < 1$}\\ 0 & \mbox{ otherwise.} \end{array}\right.$$
   
   
   Show that $X$ and $Y$ are uncorrelated but not independent.
4. Given random variables $X$ and $Y$ where the pdf of $X$ is given by
   
   $$f_{X}(x)=e^{-x}, \ \ x > 0,$$
   
   
   and $Y$ is discrete. The conditional distribution of $Y$ given X=x is given by
   
   $$f_{Y\vert X=x}(y)= \frac{e^{-x}x^y}{y!}, \ \ y=0, \ 1, \ 2, \ \ldots.$$
   
   
   Show that the marginal distribution of $Y$ is given by
   
   $$f_Y(y)=(1/2)^{y+1}, \ \ y=0, \ 1, \ 2, \ \ldots.$$
   
   

5. Use the mgf of the multinomial distribution to show that
   1. $E(X_i)=np_i$;
   2. Var $(X_i)=np_i(1-p_i)$;
   3. cov $(X_i, \; X_j)=-np_i p_j$.

6. 1. Express the following quadratic forms as ${\bf x}'{\bf A}{\bf x}$

      (i)

      $3x^2+y^2+3z^2+8xy-2xz+4yz$,

      (ii)

      $d_1x_1^2+d_2x_2^2+d_3x_3^2+d_4x_4^2$.

   2. Write the following quadratic form as ${\bf x}'{\bf A}{\bf x}$ and show that it is positive definite:
      
      $$\frac{1}{2}x_1^2 + 3x_2^2 + 3 x_3^2 + 2x_1 x_3.$$
      
      

7. If matrix A is idempotent and ${\bf A} + {\bf B} = {\bf I}$, show that B is idempotent and ${\bf A}{\bf B}={\bf B}{\bf A}={\bf0}$.
8. Given the matrix A below, find the eigenvalues and eigenvectors. Hence find the matrix P such that P$'$AP is diagonal and has as its diagonal elements, the eigenvalues of A.
   
   $$A=\left[ \begin{array}{rrr} 1& 2& 1\\ 2&1&-1\\ 1& -1& -2 \end{array}\right]$$