1. ## Bivariate transformation?

Hello, I'm having problem with a book question from statistical inference, Casella, Berger excercise 4.19 a).

X_1 and X_2 are two i.i.d R.V - following standard normal distribution.

Find the Pdf of ((X_1-X_2)^2)/2.

I guess I create a new variable called Z = ((X_1-X_2)^2)/2. But I can't use a bivariate transformation as there is only one mapping?

What I think so far is that I substitute in Z and then divide the new pdf in to two formulas, where i write in ((X_1-X_2)^2)/2 explicitly.

Sorry bad english, tried to keep it short.

Kind regards,

Talik

2. ## Re: Bivariate transformation?

You should be able to figure this out just by taking it one step at a time.

Do you know the distribution of X_1 - X_2?

3. ## Re: Bivariate transformation?

Originally Posted by Dason
You should be able to figure this out just by taking it one step at a time.

Do you know the distribution of X_1 - X_2?
Yes, if I substitute that in, I get two i.i.d std n(0,1) ?

I can write it as (1/sqrt(pie*2))*exp(((-X_1)^2)/2)*exp(((-X_1)^2)/2))

Written in bold is one standard normal.
Feels like im attacking it the wrong way.

if i use the original ((X_1-X_2)^2)/2 and root it I will get the same answer, right?
If i don't it will be chi square?

4. ## Re: Bivariate transformation?

But what distribution does their difference have? In other words.... Do you know the distribution of X_1 - X_2? (not the distribution of X_1 and X_2 seperately but what is the distribution of say Y = X_1 - X_2)

5. ## Re: Bivariate transformation?

I keep updating my same post, haha.

(1/sqrt(pie*2))*exp(((-X_1)^2)/2)*exp(((-X_1)^2)/2))
This is the distribution of the difference?

6. ## Re: Bivariate transformation?

Don't go changing posts afterwards - people don't review all the posts in a thread to make sure they haven't changed since the time they viewed the thread. I really feel like you're overcomplicating my question. I'm not asking about the pdf. I'm just saying... if you take the different of two standard normal distributions - what distribution does it have? Is it normal? Is it gamma? What are the parameters. This is definitely not the part that should be giving you troubles.

7. ## Re: Bivariate transformation?

Originally Posted by Dason
Don't go changing posts afterwards - people don't review all the posts in a thread to make sure they haven't changed since the time they viewed the thread. I really feel like you're overcomplicating my question. I'm not asking about the pdf. I'm just saying... if you take the different of two standard normal distributions - what distribution does it have? Is it normal? Is it gamma? What are the parameters. This is definitely not the part that should be giving you troubles.
I'm sorry. I'm overcomplicating because I dont understand fully.
It should be standard normal with n(0,2).

8. ## Re: Bivariate transformation?

Ok so let Y~N(0,2). We now want to find Y^2/2. Do you know what distribution you'll get if you square Y? If something was slightly different would it make it easier?

9. ## Re: Bivariate transformation?

Originally Posted by Dason
Ok so let Y~N(0,2). We now want to find Y^2/2. Do you know what distribution you'll get if you square Y? If something was slightly different would it make it easier?
I know for a fact that a standard normal distribution squared becomes chi-squared with 1 degree of freedom. So the answer is chi-squared. The thing is, i know that I can reason my way to this answer. But it I don't think it is rigoures enough?

10. ## Re: Bivariate transformation?

Well you're close. But keep in mind that if you square a STANDARD normal you get a chi-square. Y = X_1 - X_2 isn't standard normal. But it seems like you understand what you're shooting for. See if you can play around with it to get the desired results.

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