random variables - simple questions

I am having some trouble with some simple questions concerning random variables.

The first: Suppose we have a random variable X with zero mean and variance sigma^2. What is the probability distribution of Y = X^2?

Here's my argument:

	P[Y=y]
	= P[X^2 = y]
	= P[|X| = sqrt(y)]
	= P[X=sqrt(y) OR X=-sqrt(y)]
	= P[X=sqrt(y)] + P[X=-sqrt(y)]
	= 2 P[X=sqrt(y)]

We have for P[X=x] the usual normal (gaussian) distribution:

	P[X=x] = 1/(sigma Sqrt(2Pi)) Exp[-x^2/(2 sigma^2)]

so I expect that I can just substitute in x=sqrt(y):

	P[Y=y] = 2 P[X=sqrt[y]] = 2/(sigma Sqrt(2Pi)) Exp[-y/(2 sigma^2)]

But that is not right. It doesn't integrate to unity and it doesn't give the correct expectation value for Y.

Using the cumultative distribution function I follow a similar argument and get the answer:

	P[Y=y] = 1/(sigma Sqrt(2 Pi y)) Exp[-y/(2 sigma^2)]

(Notice that there is a 1/sqrt(y) in there, and no factor of 2 out front.)

I believe that answer is correct (it passes my checks)... but where does the logic fail in the argument using probability density functions?

The second question: Suppose we have independent random variables X and Y, both normally distributed with zero mean and unit variance. Define U=aX+bY and V=cX+dY. Show that U and V are independent if and only if (a,b) and (c,d) are orthogonal, i.e. ac+bd=0.

I'm not sure what to use as the test for independence here. I know <UV>=<U><V> is a <i>necessary</i> condition for the independence of U and V, but it is not in general sufficient. (Though I read somewhere that it is, in the case of normally distributed random variables??)

thanks,
Tobin

.

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