
joint / marginal pdf
Let the independent random variables X1 and X2 be N(0,1) and $\displaystyle X^{2}(r)$, respectively. Ley $\displaystyle Y1=\frac{X1}{\sqrt\frac{X^{2}}{r}}$ and Y2=X2.
a) Find the joint pdf of Y1 and Y2
b) Determine the marginal pdf of Y1 and show that Y1 has a t distribution
Thanks!

Hello,
I assume it's $\displaystyle Y_1=\frac{X_1}{\sqrt{\frac{X_2^2}{r}}}$ ?
Write down the joint pdf of $\displaystyle (X_1,X_2)$, which is just the product of their pdf (because they're independent)
Then, you can see that $\displaystyle X_1=Y_1\cdot\sqrt{\frac{X_2^2}{r}}=Y_1\cdot\sqrt{\ frac{Y_2^2}{r}}=Y_1Y_2 \cdot r^{1/2}$ (because $\displaystyle Y_2=X_2>0$)
And then make the transformation
$\displaystyle T~:~ \mathbb{R}\times \mathbb{R}_+^* \to \mathbb{R}\times \mathbb{R}_+^*$
$\displaystyle (x_1,x_2) \mapsto (y_1,y_2)=\left(\frac{x_1}{\sqrt{\frac{x_2^2}{r}}} ,x_2\right)$
Also, don't forget to divide by the Jacobian !
that's pretty "long" to do, so it'd be good for you and for us that you show what you've tried/done, what you want us to correct.

This is okay. At first I thought you had two indep st. normals.
You need this $\displaystyle X_2$ to be a $\displaystyle \chi^2_r$.
I was lost on that missing subscript at first.
The joint density of Y1 and Y2 should factor giving you independence.

I'm sorry. It is supposed to be $\displaystyle Y_{1}=\frac{X_{1}}{\sqrt\frac{X_{2}}{r}} $and I haven't a clue where to begin. Thanks

1 Attachment(s)
Is this anywhere near what I am supposed to do?
Thanks!
Attachment 11650

(3) is way off. you need to review jacobians from calculus 3.

Thank you. I just have my statistics book though. Could you help point me in the right direction? I don't know what I did wrong based on my text.

You should reread what Moo told you.
Start with..Write down the joint pdf of http://www.mathhelpforum.com/mathhe...6bc02d441.gif, which is just the product of their pdf ..
then obtain the jacobian and change variables.
Here's a review that explains the change for polar, but it's the same idea.
http://www.maths.abdn.ac.uk/~igc/tch...es/node77.html
Here's some more calc 3...
http://tutorial.math.lamar.edu/Class...Variables.aspx
And this should be outlined in you stat book too.