Thread: Simple if not fot the Trick!

If it is already given that X1 and X2 are independent R.V. each having p.d.

f(x) = { e^-x , x > 0
{ 0 , otherwise

How would one go about showing that R.V. (s) R.V.s Y1 and Y2 are independent when Y1 = X1 + X2, Y2 =

Do I need the abstraction of Y1 and Y2 or can I just leave it in terms of X? I'm not sure if thats the right question. The additional R.V.s are throwing me off.

Oh, and can someone give me a link to a tutorial or listing of the Math Markup used to display Math notation so my next thread or answer can look better.

Thank you

I don't know what this means...Do I need the abstraction of Y1 and Y2 or can I just leave it in terms of X?

You need to make a two dimensional change.
It's just calc 3.
The joint density of X_1 and X_2 is the product of their marginals.
So, f(x_1,x_2)=exp{-(x_1+x_2)} in the irst quadrant, x_1>0, x_2>0.
NOW ge the jacobian, which you should recall is the determinant of the four partial derivative.
INSERT your new rvs, y_1 and y_2 and multiple by the absolute value of your jacobian and you're done.
Clearly, since X_1 and X_2 are exponentials, which are gamma (1,1)
We have via MGFs that the sum is a gamma (2,1)
And one can see that Y_2 is between 0 and 1, hence it's Beta with mean .5.
I think it's a Beta(1,1).
But to see that they are independent you need the joint density of Y_1 and Y_2.


, just learning your TeX here

Originally Posted by Haskell

...
Oh, and can someone give me a link to a tutorial or listing of the Math Markup used to display Math notation so my next thread or answer can look better.

Thank you

See the tutorial here:

http://www.mathhelpforum.com/math-he...-tutorial.html

Math Help has an entire forum devoted to LaTex:

http://www.mathhelpforum.com/math-help/latex-help/

Also, you can see other user's LaTex by simply pressing the Quote button at the end of a post.