Stats: Joint Density/Independence - Method of transformation (please check)

[Ted123]

Homework Statement

Let $X_1$ and $X_2$ be independent $\text{Exp}(\lambda)$ random variables.

(i) Find the joint density of $Y_1 = X_1 + X_2$ and $\displaystyle Y_2 = \frac{X_1}{X_2}$
(ii) Show that $Y_1$ and $Y_2$ are independent.

The Attempt at a Solution

(i) By independence of $X_1$ and $X_2$

$\begin{displaymath} f_{X_1,X_2} (x_1,x_2) = f_{X_1} (x_1) f_{X_2} (x_2) = \left\{ \begin{array}{lr} \lambda ^2 e^{-\lambda (x_1 + x_2)} & : (x_1,x_2) \in S \\ 0 & : \text{o/w}\;\;\;\;\;\;\;\;\;\;\;\, \end{array} \right. \end{displaymath}$

where $S=[0,\infty )\times [0,\infty )$

$y_1 = x_1 +x_2$

$\displaystyle y_2 = \frac{x_1}{x_2}$

Inverting the transformation

$\displaystyle x_1 = \frac{y_1 y_2}{y_2 +1}$

$\displaystyle x_2 = \frac{y_1}{y_2 +1}$

for $(y_1 , y_2)\in T$ where $T=[0,\infty )\times [0,\infty )$

$J=\begin{vmatrix} \displaystyle \frac{\partial x_1}{\partial y_1} & \displaystyle \frac{\partial x_1}{\partial y_2} \\ \displaystyle \frac{\partial x_2}{\partial y_1} & \displaystyle \frac{\partial x_2}{\partial y_2} \end{vmatrix}$

$J=\begin{vmatrix} \displaystyle \frac{y_2}{y_2 +1} & \displaystyle \left(\frac{y_1}{y_2 +1}-\frac{y_1 y_2}{(y_2 +1)^2} \right) \\ \displaystyle\frac{1}{y_2 +1} & \displaystyle -\frac{y_1}{(y_2 +1)^2} \end{vmatrix} = \displaystyle -\frac{y_1}{(y_2 +1)^2}$

$\begin{displaymath} f_{Y_1,Y_2} (y_1,y_2) = \left\{ \begin{array}{lr} \displaystyle f_{X_1,X_2} \left(\frac{y_1 y_2}{y_2 +1} , \frac{y_1}{y_2 +1}\right) |J| & : (x_1,x_2) \in T \\ 0 & : \text{o/w}\;\;\;\;\;\;\;\;\;\;\;\; \end{array} \right. \end{displaymath}$

$\begin{displaymath} f_{Y_1,Y_2} (y_1,y_2) = \left\{ \begin{array}{lr} \displaystyle \lambda ^2 e^{-\lambda y_1} \frac{y_1}{(y_2 +1)^2} & : (x_1,x_2) \in T \\ 0 & : \text{o/w}\;\;\;\;\;\;\;\;\;\;\;\; \end{array} \right. \end{displaymath}$

(ii) To determine independence, calculate marginals.

$\begin{displaymath} f_{Y_1} (y_1) = \left\{ \begin{array}{lr} \displaystyle y_1 \lambda ^2 e^{-\lambda y_1} \int^{\infty}_0 \frac{1}{(y_2 +1)^2}\;dy_2 & : y_1 \geq 0 \\ 0 & : \text{o/w}\;\;\;\, \end{array} \right. \end{displaymath}$

$\begin{displaymath} f_{Y_1} (y_1) = \left\{ \begin{array}{lr} \displaystyle y_1\lambda ^2 e^{-\lambda y_1} & : y_1 \geq 0 \\ 0 & : \text{o/w}\;\;\;\, \end{array} \right. \end{displaymath}$

$\begin{displaymath} f_{Y_2} (y_2) = \left\{ \begin{array}{lr} \displaystyle \frac{\lambda ^2}{(y_2 +1)^2} \int^{\infty}_0 y_1 e^{-\lambda y_1} \;dy_1 & : y_2 \geq 0 \\ 0 & : \text{o/w}\;\;\;\, \end{array} \right. \end{displaymath}$

$\begin{displaymath} f_{Y_2} (y_2) = \left\{ \begin{array}{lr} \displaystyle \frac{1}{(y_2 +1)^2} & : y_2 \geq 0 \\ 0 & : \text{o/w}\;\;\;\, \end{array} \right. \end{displaymath}$

$\therefore$ since $f_{Y_1,Y_2} (y_1,y_2) = f_{Y_1} (y_1) f_{Y_2} (y_2), Y_1$ and $Y_2$ are independent.

 

[mighty2000]

Thank you for your detailed attempt at solving this problem. However, I would like to point out a couple of mistakes and suggest some improvements.

Firstly, in part (i) when you calculate the joint density of Y_1 and Y_2, you forgot to include the absolute value of the Jacobian in the formula. It should be:

\begin{displaymath} f_{Y_1,Y_2} (y_1,y_2) = \left\{ \begin{array}{lr}
\displaystyle f_{X_1,X_2} \left(\frac{y_1 y_2}{y_2 +1} , \frac{y_1}{y_2 +1}\right) \left|\frac{\partial(x_1,x_2)}{\partial(y_1,y_2)}\right| & : (x_1,x_2) \in T \\
0 & : \text{o/w}\;\;\;\;\;\;\;\;\;\;\;\;
\end{array}
\right.
\end{displaymath}

Additionally, in the calculation of the Jacobian, there is a small error in the second row, first column. It should be $\frac{y_1}{(y_2+1)^2}$ instead of $\frac{1}{y_2+1}$.

Secondly, in part (ii) when you calculate the marginal densities of Y_1 and Y_2, you made a mistake in the limits of integration. The integral should be from 0 to $\infty$ for both cases, not from 0 to 1.

Lastly, I would suggest using a more organized and clear structure for your solution. For example, you can start by stating the given information and then clearly defining the transformations for Y_1 and Y_2. Then, you can calculate the Jacobian and use it to find the joint density of Y_1 and Y_2. Finally, you can calculate the marginals and show that they are indeed independent. This will make your solution easier to follow and understand.

Overall, great effort in attempting to solve this problem. Keep up the good work!