Hello,
See here : http://www.well.ox.ac.uk/~valdar/reference/jacobian.pdf
I have a problem and I am at a loss:
Suppose that Z is a standard normal random variable and Y1 and Y2 are χ2 distributed random variable with υ1 and υ2 degrees of freedom, respectively. Further, assume that Z and Y1 and Y2 are independent.
a) Define W = (Z/SQRT(Y1)). Find E(W) and V(W). What assumptions do you need about the value of υ1?
b) Define U= Y1/Y2 . Find E(U) and V(U). What assumptions do you need about the value of υ1 and υ2?
I know the moment generating functions of Z, Y1 & Y2, but I don't know how I am supposed to find the distribution of W considering we are dividing densities... Any help would be vastly appreciated!
Hello,
See here : http://www.well.ox.ac.uk/~valdar/reference/jacobian.pdf
df=degrees of freedom
I figured you were given this problem because you're studying the F and T distributions.
There are three ways to attack this.
1 Derive the densities, which is messy and unnecessary given that you're only asked to obtain the moments.
2 Find the moments via the Y's, which isn't that hard
3 Use the F and T moments and obtain the moments of these two in a minute.
Using the last one...
So
and
Now look up the mean and variance of an F, which isn't that hard to derive...
(thats my second suggested technique)
http://en.wikipedia.org/wiki/F-distribution
where and isn't that hard to obtain either.