How do we identify the distribution of a pivotal quantity? I have been given a pivotal quantity of $2\beta\sum_{i=1}^4X_i$ to determine a confidence interval of random sample $\underline{X}=(X_1,...,X_4)$ from a $\Gamma(4,\beta)$ distribution.
Initially, I want to find the distribution of this pivotal quantity, and why it can be used to construct a confidence interval for $\beta$.
I understand that provided the quantity is a function of the observations and parameter, in this case $g(\underline{x};\beta)$, and the distribution is known and independence of $\beta$ holds, then it can be used as a pivotal quantity. However, I am lost as how to systematically approach such a pivotal quantity to determine its distribution.
I have tried to provide an argument concerning the fact that each $X_i\sim\Gamma(4,\beta)$ and as such $\sum_{i=1}^4X_i\sim\Gamma(16,\beta)$ but this has been to no avail - as I cannot find independence of $\beta$.
Any and all help would be much appreciated. Thank you in advance.
 A: We start our answer by denoting the pivotal quantity by $Y_i=2 \beta X_i$.
As above, each $X_i \sim Gamma(4,\beta)$, so we can obtain that the probability density function (pdf) of $X$ is $f_X(x)=\frac{\beta^4}{6}x^3\exp(-\beta x)$.
We then apply the transformation process, beginning with the cumulative distribution function, $F_Y(y)=P(Y \leq y)=P(2 \beta X \leq Y)=P(X\leq\frac{y}{2\beta})=F_X(\frac{y}{2\beta})$. Applying to densities, we obtain: $f_Y(y)=F_Y'(y)=F_X'(\frac{y}{2\beta})\times\frac{1}{2\beta}=f_X(x)\times\frac{1}{2\beta}=\frac{y^3}{96}\exp(-\frac{y}{2})\textbf{1}_{y>0}$.
Noting that the generic pdf of a $\chi_n^2$ distribution is $\frac{1}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}x^{\frac{n}{2}-1}e^{-\frac{x}{2}}$. Selecting $n=8$ degrees of freedom allows us to obtain $f_Y(y)$.
We note that $\chi_8^2$ is the distribution for one $X_i$. Therefore, $2\beta\sum_{i=1}^4X_i=\sum_{i=1}^4Y_i$ gives a distribution of $\chi_{32}^2$, by the properties of the distribution.
As above, this is a valid result as $\chi_{32}^2$ is independent of $\beta$ and consists of observations $\underline{X}$.
