ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2338 on D3838: Gamma random positive real number
Finite sum of I.I.D. exponential random positive real numbers is a gamma random random positive real number
Formulation 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 D2713: Independent random collection
Then \begin{equation} \sum_{n = 1}^N T_n \overset{d}{=} \text{Gamma}(N, \theta) \end{equation}
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 D2713: Independent random collection