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Is there a well-known reference in statistics to cite that state the PDF of a gamma random variable $$f(x)= \frac{1}{\Gamma{(k) \theta^k}} x^{k-1} e^{-\frac{x}{\theta}}$$

with $\text{mean}= k \theta$ and $\text{variance} = k \theta^2$ also a reference for random variable transformation for the PDF and CDF transformation of variables, for example this relation:

$$f_Y(y) = f_X(g^{-1}(y)) \left| \frac{\mathrm d}{\mathrm dy} g^{-1}(y) \right|$$

I am searching for a well known and cited references for these topics.

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  • $\begingroup$ That is two different questions. Wikipedia points to references on the gamma distribution and on changes of variables $\endgroup$ Commented Dec 5, 2023 at 20:52

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Casella, G., & Berger, R. L. (2002). Statistical Inference. 2nd ed. Australia ; Pacific Grove, CA, Thomson Learning.

The equation $$f_Y(y)=f_X(g^{-1}(y))\bigl|\frac{d}{dy}g^{-1}(y)\bigr|$$ for $y\in \mathcal{Y}$, where $\mathcal{X}=\{x:f_X(x)>0\}$ and $\mathcal{Y}=\{y:y=x(x)\text{\for some\ }x\in \mathcal{X}\}$ is equation $2.1.10$ in Theorem $2.1.5$ (p $51$).

The pdf of the gamma distribution is given in equation $3.3.6$ (p $99$) and the mean and variance are given on page $100$.

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