ThmDex – An index of mathematical definitions, results, and conjectures.
Finite sum of uncorrelated identically distributed exponential random positive real numbers is a gamma random random positive real number

Let $T_1, \ldots, T_N \in \text{Exponential}(\theta)$ each be an D214: Exponential random positive real number such that
 (i) $T_1, \ldots, T_N$ is an D3842: Uncorrelated random collection
Then $$\sum_{n = 1}^N T_n \overset{d}{=} \text{Gamma}(N, \theta)$$
Subresults
 ▶ R5251: Exponential random positive real number is a gamma random positive real number ▶ R2338: Finite sum of I.I.D. exponential random positive real numbers is a gamma random random positive real number
Proofs
Proof 0
Let $T_1, \ldots, T_N \in \text{Exponential}(\theta)$ each be an D214: Exponential random positive real number such that
 (i) $T_1, \ldots, T_N$ is an D3842: Uncorrelated random collection
Using results
we have $$\begin{split} \mathbb{E}(e^{i t \sum_{n = 1}^N T_n}) = \prod_{n = 1}^N \mathbb{E}(e^{i t T_n}) & = \prod_{n = 1}^N \frac{\theta}{\theta - i t} \\ & = \prod_{n = 1}^N \frac{\theta}{\theta - i t} \frac{1 / \theta}{1 / \theta} = \prod_{n = 1}^N \frac{1}{1 - i t / \theta} = \frac{1}{(1 - i t / \theta)^N} \end{split}$$ By definition, a gamma random positive real number with parameters $N$ and $\theta$ has characteristic function $(1 - i t / \theta)^{- N}$, so we are done. $\square$