Sums of Independent Random Variables

[jrk012]

Homework Statement

So, I know the pdf for independent random variables is found by using the convolution; the pdf is f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy, but can I just use the density function for a function of a random variable instead; that is: f[sub:X+Y](x[u,v], y[u,v])(Jacobian Inverse(x[u,v], y[u,v])) and then integrate it? It seems much easier that way.

Homework Equations

f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy

f[sub:X+Y](x[u,v], y[u,v])(Jacobian Inverse(x[u,v], y[u,v]))

Jacobian Determinant: (∂u/∂x)(∂v/∂y)-(∂u/∂y)(∂v/∂x)

The Attempt at a Solution

More of a question on coursework than homework or a specific problem. However when I find the density function when U=X+Y I get f(x[u-v], y[v])

 

[Ray Vickson]

Homework Statement

So, I know the pdf for independent random variables is found by using the convolution; the pdf is f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy, but can I just use the density function for a function of a random variable instead; that is: f[sub:X+Y](x[u,v], y[u,v])(Jacobian Inverse(x[u,v], y[u,v])) and then integrate it? It seems much easier that way.

Homework Equations

f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy

f[sub:X+Y](x[u,v], y[u,v])(Jacobian Inverse(x[u,v], y[u,v]))

Jacobian Determinant: (∂u/∂x)(∂v/∂y)-(∂u/∂y)(∂v/∂x)

The Attempt at a Solution

More of a question on coursework than homework or a specific problem. However when I find the density function when U=X+Y I get f(x[u-v], y[v])

So U = X+Y, but what is V?

RGV

 

[jrk012]

They said "Hint: V=Y" so I wasn't sure if that was supposed to be assumed known or if there were different ways to do it. Sorry I forgot to put that in there!