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My question is very similar to the ones asked before (I have looked at all of them on Cross-Validated) but it is more about house-keeping and making sure it matches precisely transformation theorems conditions as they are described in Chapter 2 of Casella and Berger (pages 47-53, theorems of interest are 2.1.3., 2.1.5, and 2.1.8). So the main question is,

  1. Which exact transformation to use and I need to make sure that assumptions are met exactly as they are described in the book;

  2. I am also puzzled about any restrictions on the parameters that I might need to include.

So here is the problem and my solution:

We need to convert gamma into chi-squared in order to be able to use a chi-squared table. $X = \frac{2W}{\beta}, $ where W is a sum of gammas.

If $W \approx \textrm{Gamma}(20, 1/3)$ we need to transform it into $ \chi^2_{(2n\alpha)}$ and the answer is, since we are essentially interested in the degrees of freedom of this $\chi^2$ distribution, the dfs=40 (the answer).

So here is my take on it: At first I am finding an mgf using the property of the sums of Gammas from Chapter 4 of Casella and Berger,page 183, example 4.6.8 "Mgf of a sum of gamma variables":

$$E[e^{tw}] = E\left[e^{t*\frac{2}{\beta}W}\right] = E\left[e^{(\frac{2}{\beta}t)W}\right] = M_w\left(\frac{2}{\beta}t\right) = \frac{1}{1- 2t} \to \chi^2_{2\alpha}$$

  • so now that I finally got this result I have a pretty silly question # 1: do I just plug in 20 into alpha to get 2*20=40 degrees of freedom result?

  • Question #2: although I feel excited about finding that property in Chapter 4 but I would still like to learn how to apply the transformation and I do not understand how to do this with transformation and more importantly, which specific transformation in Casella and Berger I should use out of all those I have listed and how it meets the conditions: so I suspect it's Theorem 2.1.5, so I will need to know if the function is monotone, how do I know it's monotone (I have taken calculus more than 15 years ago) and also, do I need to worry about on which intervals the partition will be increasing/decreasing?

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