(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose we have a function, f(x,y) = e^-x * e^-y , 0<=x< ∞, 0<=y<∞,

where X and Y are exponential random variables with mean = 1. (For those who may not know, all this means is ∫(x*e^(-x) dx) from 0 to ∞ = 1, and the same for y)

Suppose we want to transform f(x,y) into f(z), where the transformation is Z = X-Y. Find f(z)

2. Relevant equations

f(x,y) = e^-x * e^-y , 0<=x< ∞, 0<=y<∞

Z = X-Y

3. The attempt at a solution

So I decided to transform -Y into W. So we have -Y<=W, which implies Y>=-W

∫e^-y dy from -w to ∞...

-e^-y from -w to ∞

0 + e^-(-w)

e^w

Differentiate wrt w

f(w) = e^w -∞ < w <= 0

So now we have Z = X+W

Z-W = X

We'll just let W stay as is for this problem.

The jacobian of this transformation is 1.

So we have f(z) = ∫e^-(z-w)*e^w dw from -∞ to z,

This becomes ∫e^-z*e^2w dw from -∞ to z

This becomes e^-z * (e^2w)/2 from -∞ to z

This becomes e^-z * ((e^2z)-0)/2

Which becomes (e^z)/2

The domain of z is -∞ to ∞, however, this integral does not evaluate to 1. As a matter of fact, it does not even converge.

Any help would be greatly appreciated!!

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# Transformation of Random Variables (Z = X-Y)

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