t distribution

**noob mathematician** · August 21st 2009, 10:52 PM

It was given that a random variable T is said to have a tn distribution when:

$T=\frac{Z}{\sqrt{U/n}}$
but while proving for the density function (pdf) of a tn distribution, why is is equal to the joint distribution $\int_{0}^{\infty}f_{T,U}(t,u).du$ ?

**matheagle** · August 21st 2009, 10:59 PM

You first find the joint denisty of Z and U, where you needed to say that

$U\sim \chi^2_n$ and $Z\sim N(0,1)$ (and they are indep.)

Then pick a dummy varable, say W=Z or W=U.
(IF you let W=U, then you can do this, but you better check your bounds of integration.)

Use calc3 to find the density of T and W.

THEN integrate out the W and you have the density of U.

**noob mathematician** · August 21st 2009, 11:36 PM

Well initially I was trying to solve it by 1st letting:
$Y=\sqrt{U/n}$
$f_Y(y)=f_u(ny^2)(2ny)$

then follow by:
$T=\frac{Z}{Y}$
$f_T(t)=\int_{-\infty}^{\infty}|y|f_Y(y)f_z(yt).dy$

but it turned out to be very complicated

**matheagle** · August 22nd 2009, 06:26 AM

this is in many books
I bet you can find it online too.
I did a quick search yesterday, but didn't find it.
I have it some where in my office and I know I can do it,
but I don't have the time.
It's in Hogg and Craig.

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