Question

Let Xi, X2, X3 be i.id. N(0.1) Suppose Yı = Xi + X2 + X3,Ý, = Xi-X2, у,-X,-X3. Find the joint pdf of Y-(y, Ya, y), using: andom variables. a. The method of variable transformations (Jacobian), b. Multivariate normal distribution properties.

Answer 1

$a.~The~joint~distribution~of~(X_1,X_2,X_3)~is\\ f(x_1,x_2,x_3)=\frac{1}{(2\pi)^{3/2}}\exp\left(-\frac{x_1^2&plus;x_2^2&plus;x_3^2}{2} \right );~-\infty<x_1,x_2,x_3<\infty\\ Transform~(X_1,X_2,X_3)\rightarrow (Y_1,Y_2,Y_3)~where\\ y_1=x_1&plus;x_2&plus;x_3,~y_2=x_1-x_2,~y_3=x_1-x_3\\$

$then~x_1=\frac{y_1&plus;y_2&plus;y_3}{3},~x_2=\frac{y_1&plus;y_3-2y_2}{3},~x_3=\frac{y_1&plus;y_2-2y_3}{3};\\ J\left(\frac{x_1,x_2,x_3}{y_1,y_2,y_3}\right)=\begin{vmatrix} \frac{\partial x_1}{\partial y_1} & \frac{\partial x_1}{\partial y_2}&\frac{\partial x_1}{\partial y_3} \\ \frac{\partial x_2}{\partial y_1} & \frac{\partial x_2}{\partial y_2}&\frac{\partial x_2}{\partial y_3} \\ \frac{\partial x_3}{\partial y_1} & \frac{\partial x_3}{\partial y_2}&\frac{\partial x_3}{\partial y_3} \end{vmatrix}=\begin{vmatrix} 1/3 & 1/3 & 1/3\\ 1/3& -2/3 &1/3 \\ 1/3& 1/3 &-2/3 \end{vmatrix}=\frac{1}{3},\\ ~|Jacobian|=1/3\\ Now,~\sum_{i=1}^3x_i^2=3y_1^2&plus;6y_2^2&plus;6y_3^2-6y_2y_3\\$

$Hence~joint~distribution~of~Y_1,Y_2,Y_3~is\\ \frac{1}{(2\pi)^{3/2}\times 3}\exp\left(-\frac{1}{6}(3y_1^2&plus;6y_2^2&plus;6y_3^2-6y_2y_3)\right)\\ =\frac{1}{(2\pi)^{3/2}\times 3}\exp\left(-\frac{1}{2}(y_1^2&plus;2y_2^2&plus;2y_3^2-2y_2y_3)\right);~-\infty<y_1,y_2,y_3<\infty$

$b.\\~\begin{pmatrix} Y_1\\ Y_2\\ Y_3 \end{pmatrix}=\begin{pmatrix} 1 &1 &1 \\ 1&-1 &0 \\ 1& 0& -1 \end{pmatrix}\begin{pmatrix} X_1\\ X_2\\ X_3 \end{pmatrix}={\bf A}\begin{pmatrix} X_1\\ X_2\\ X_3 \end{pmatrix}\sim N_3({\bf A}{\bf 0},~{\bf A}{\bf A}^\prime)\\ where,~\begin{pmatrix} X_1\\ X_2\\ X_3 \end{pmatrix}\sim N({\bf 0},~{\bf 0}=null~vector,~{\bf I_3}),\\~{\bf I}_3=identity ~matrix~order~3, {\bf A}=\begin{pmatrix} 1 &1 &1 \\ 1&-1 &0 \\ 1& 0& -1 \end{pmatrix}.\\ {\bf A}{\bf 0}={\bf 0},~{\bf A}{\bf A}^\prime=\begin{pmatrix} 3 &0 & 0\\ 0 &2 &1 \\ 0 & 1& 2 \end{pmatrix}$