- #1
cpatel23
- 16
- 0
Before I begin, here is the question:
If the PDF of two independent random variables X and Y are:
f(x) = exp(-x)u(x)
f(y) = exp(-y)u(y)
Determine the join probability density function (JPDF) of Z&W defined by:
Z = X+Y
W = X/(X+Y).
So, I know how to solve this except for one thing. How do I get the expression for Z and W.
For Z do I literally just add f(x) + f(y) meaning z = exp(-x) + exp(-y) for (x,y) >0? Same with W?
Once I get the expressions I just find fxy(x,y) and divide by the determinant of the Jacobian. The problem is that the Jacobian depends on the derivative of Z and W which I do not know how to get an expression for.
Please help.
If the PDF of two independent random variables X and Y are:
f(x) = exp(-x)u(x)
f(y) = exp(-y)u(y)
Determine the join probability density function (JPDF) of Z&W defined by:
Z = X+Y
W = X/(X+Y).
So, I know how to solve this except for one thing. How do I get the expression for Z and W.
For Z do I literally just add f(x) + f(y) meaning z = exp(-x) + exp(-y) for (x,y) >0? Same with W?
Once I get the expressions I just find fxy(x,y) and divide by the determinant of the Jacobian. The problem is that the Jacobian depends on the derivative of Z and W which I do not know how to get an expression for.
Please help.