joint / marginal pdf

**jennifer1004** · May 18th 2009, 09:11 AM

Let the independent random variables X1 and X2 be N(0,1) and $X^{2}(r)$ , respectively. Ley $Y1=\frac{X1}{\sqrt\frac{X^{2}}{r}}$ 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!

**Moo** · May 18th 2009, 09:31 AM

Hello,

I assume it's $Y_1=\frac{X_1}{\sqrt{\frac{X_2^2}{r}}}$ ?

Write down the joint pdf of $(X_1,X_2)$ , which is just the product of their pdf (because they're independent)

Then, you can see that $X_1=Y_1\cdot\sqrt{\frac{X_2^2}{r}}=Y_1\cdot\sqrt{\ frac{Y_2^2}{r}}=Y_1Y_2 \cdot r^{-1/2}$ (because $Y_2=X_2>0$ )

And then make the transformation
$T~:~ \mathbb{R}\times \mathbb{R}_+^* \to \mathbb{R}\times \mathbb{R}_+^*$
$(x_1,x_2) \mapsto (y_1,y_2)=\left(\frac{x_1}{\sqrt{\frac{x_2^2}{r}}} ,x_2\right)$

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.

**matheagle** · May 18th 2009, 02:35 PM

This is okay. At first I thought you had two indep st. normals.

You need this $X_2$ to be a $\chi^2_r$ .

I was lost on that missing subscript at first.
The joint density of Y1 and Y2 should factor giving you independence.

**jennifer1004** · May 20th 2009, 06:55 PM

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

**jennifer1004** · May 27th 2009, 05:18 PM

Is this anywhere near what I am supposed to do?
Thanks!
Stats.pdf

**matheagle** · May 27th 2009, 05:38 PM

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

**jennifer1004** · May 27th 2009, 05:41 PM

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.

**matheagle** · May 27th 2009, 05:48 PM

You should reread what Moo told you.
Start with..Write down the joint pdf of

, 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.

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