I've learned sum of exponential random variables follows Gamma distribution.
But everywhere I read the parametrization is different. For instance, Wiki describes the relationship, but don't say what their parameters actually mean? Shape, scale, rate, 1/rate?
Exponential distribution: $x$~$exp(\lambda)$
$$f(x|\lambda )=\lambda {{e}^{-\lambda x}}$$
$$E[x]=1/ \lambda$$
$$var(x)=1/{{\lambda}^2}$$
Gamma distribution: $\Gamma(\text{shape}=\alpha, \text{scale}=\beta)$
$$f(x|\alpha ,\beta )=\frac{1}{{{\beta }^{\alpha }}}\frac{1}{\Gamma (\alpha )}{{x}^{\alpha -1}}{{e}^{-\frac{x}{\beta }}}$$
$$E[x]=\alpha\beta$$
$$var[x]=\alpha{\beta}^{2}$$
In this setting, what is $\sum\limits_{i=1}^{n}{{{x}_{i}}}$? What would the correct parametrization be? How about extending this to chi-square?
Best Answer
The sum of $n$ independent Gamma random variables $\sim \Gamma(t_i, \lambda)$ is a Gamma random variable $\sim \Gamma\left(\sum_i t_i, \lambda\right)$. It does not matter what the second parameter means (scale or inverse of scale) as long as all $n$ random variable have the same second parameter. This idea extends readily to $\chi^2$ random variables which are a special case of Gamma random variables.