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November 8th, 2014, 06:29 AM  #1 
Newbie Joined: Nov 2014 From: Brussels Posts: 2 Thanks: 0  Transformation of a Random Variable (chisquare)
We have a random variable x with p.d.f. $\sqrt{\dfrac{\theta}{\pi x}}\exp(x\theta)$, x>0 and θ a positive parameter. We are required to show that 2θx has a χ2 distribution with 1 degree of freedom and deduce that, if $x_1,\dots,x_n$ are independent r.v. with this p.d.f., then $2\theta\sum_{i=1}^n x_i$ has a χ2 distribution with n degrees of freedom. Using transformation y=2θx I found the pdf of $$y=\frac{1}{\sqrt{2\pi}}y^{1/2}e^{y/2}.$$ How do I find the distribution of $2\theta\sum_{i=1}^n x_i$? Do I need to find the likelihood function (which contains $\sum_{i=1}^n x_i$) first? How do I recognise the degrees of freedom of this distribution (Is it n because it involves $x_1,\dots,x_n$, i.e. n random variables? 
November 8th, 2014, 08:07 AM  #2  
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  Quote:
Otherwise, use the moment generating function of the characteristic function. CB Last edited by CaptainBlack; November 8th, 2014 at 08:18 AM.  
November 8th, 2014, 10:38 AM  #3 
Newbie Joined: Nov 2014 From: Brussels Posts: 2 Thanks: 0 
Thank you very much for your response. Could you, please, elaborate, on how using the moment generating function would help us in this respect ( i.e. finding the 2θΣx(i=1 to n) distribution?

November 8th, 2014, 12:12 PM  #4  
Senior Member Joined: Jan 2012 From: Erewhon Posts: 245 Thanks: 112  Quote:
CB  

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chisquare, distribution, hypothesis testing, random, statistics, transformation, transformations, variable 
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