transforming univariate normal distribution?

Question

X1, X2 independent N(0,1) variables.

Deduce the distribution of:

Y = 1/2 [ (X2 - X1)^2]

Thank you for your time

Answer 1

Well, since X2 and X1 are N(0,1) any linear combination of the two is also N(0,1) so that let Z = X2 - X1 is N(0,1)
and we also know that any standard normal random variable SQUARED is Chi-squared(1) distributed that is it is distributed Chi-squred with one degree of freedom since we are only squaring one standard normal random variable.
Hence Z^2 is Chi-squared(1)
So, I conclude that Y = 1/2*[ (X2 - X1)^2]
= 1/2*[Z^2] where Z is N(0,1)
Hence Y is Chi-squared(1) distributed

Answer 2

I don't know much about probability, but I believe that the distribution of Y is:

P(Y0. In this case, the region in question is:

{(x,y): (y-x)^2<2*t}

We can describe this region most easily with the coordinates: (s,w) -> s*(1/sqrt(2))(1,1) + w*(1/sqrt(2))(-1,1). In these coordinates the region is {(s,w): w^2