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?
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
Last edited by Talik; 01-11-2016 at 07:08 PM.
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?
I don't have emotions and sometimes that makes me very sad.
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?
Last edited by Talik; 01-11-2016 at 07:24 PM.
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)
I don't have emotions and sometimes that makes me very sad.
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?
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 don't have emotions and sometimes that makes me very sad.
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 don't have emotions and sometimes that makes me very sad.
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.
I don't have emotions and sometimes that makes me very sad.
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