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If I have $X_{1},\ldots,X_{n}$ a random sample from a Weibull distribution $X\sim WEI(\theta,2)$.How can I show that $Q=2\sum\limits_{i=1}^n X_{i}^2/\theta^2\sim \chi^2(2n)$.

I have not learnt any transformations for Weibull distributions. I believe that if it has a squared term is because it got to be standar normal somehow and then became a chi-squared. The pdf of Weibull is similar to the exponential one, but that did not help. I also try to use the Jacobian to make the transformation but that sum stopped me.

Lotte
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1 Answers1

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First of all, you posted this same question earlier and it was closed because you didn't show what you tried. Adding a little paragraph about your thoughts on the question doesn't really address this. You also didn't tag the question as homework.

No. The standard normal does not play a role here.

  1. What is the CDF of the Weibull distribution with scale parameter $\theta$ and shape parameter $2$?
  2. So if $X \sim {\rm Weibull}(\theta,2)$, what is the probability $\Pr[(X/\theta)^2 \le x]$? Note that since $X$ is already a nonnegative random variable, this is simply $\Pr[X \le \theta\sqrt{x}]$.
  3. From the above, what can you conclude about the resulting distribution of each $(X_i/\theta)^2$? Does it depend on the parameter $\theta$? Do you recognize the distribution?
  4. What do you remember about the sum of $n$ IID such random variables? What is the PDF of $Q/2$? Then, what is the PDF of $Q$ itself?
  5. What is the PDF of the chi-squared distribution with $2n$ degrees of freedom?

When you have done this and shown your effort--that means showing us actual calculations, then you might get more of a response.

heropup
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  • If you were able to understand my numbered list of things to do, then you obviously have the requisite mathematical knowledge. So if you haven't tried anything, then it isn't because you don't know how, but because you don't know what to try. And if you don't know what to try, then that means you have more serious deficiencies that will not be resolved by coming here to get advice on your homework. You should figure out why you don't seem to be able to draw upon earlier, more elementary concepts. – heropup Apr 19 '14 at 21:41