Solving Joint Convolution for P(X,Y) - 65 Characters

[yamdizzle]

I solved majority of the question I just need to find the last joint density. Found the equations at part 3.

Homework Statement

Show P(X-Y=z ,Y=y) = P(X) = P(|Y|)
I showed P(X) = P(|Y|)

Homework Equations

The Attempt at a Solution

P(X=x,Y=y) = $\frac{2*(2x-y)}{\sqrt{2πT^3σ^6}}$ * exp((($\frac{-(2x-y)^2}{(2σ^2T)}$))
P(Y=y) = NormalPDF(0,Tσ^2)
P(X=x) = 2*NormalPDF(0,Tσ^2)

I don't really want to find the convolution then the Jacobian unless I have to. If there is an easier way please let me know.

 

[yamdizzle]

I took a step ahead and said:

P(X=x, Y=y) = P(X-Y=x-y , Y=y) = P(Z=z, Y=y) but I don't seem to get the right distribution.

 

[yamdizzle]

I solved majority of the question I just need to find the last joint density. Found the equations at part 3.

Homework Statement

Show P(X-Y=z ,Y=y) = P(X) = P(|Y|)
I showed P(X) = P(|Y|)

Homework Equations

The Attempt at a Solution

P(X=x,Y=y) = $\frac{2*(2x-y)}{\sqrt{2πT^3σ^6}}$ * exp((($\frac{-(2x-y)^2}{(2σ^2T)}$))
P(Y=y) = NormalPDF(0,Tσ^2)
P(X=x) = 2*NormalPDF(0,Tσ^2)

I don't really want to find the convolution then the Jacobian unless I have to. If there is an easier way please let me know.

Ignore the question There was a typo at the question. I just solved it Thanks