IF you're trying to obtain the density of a T
you need the joint density of the normal and the chi-square.
The T is a ratio of these.
You can't obtain the joint from the two marginals.
U follows a chi-square distribution with n degrees of freedom, and V = sqrt(U/n). I'm trying to get the density of V (part one in deriving the density function of the t distribution). I've tried F(V) = P(V<=v) = P(sqrt(U/n)<=v) = P(U<=n*v^2), then f(V) = d/dv(F(v)) = 2nv*f(n*v^2), then substituted n*v^2 in for x for the pdf of a chi square, but I feel like I'm doing something wrong. Can anyone give me a nudge in the correct direction?