ThmDex – An index of mathematical definitions, results, and conjectures.

Result R2338 on D3838: Gamma random positive real number
Subresult of R5243: Finite sum of uncorrelated identically distributed exponential random positive real numbers is a gamma random 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

This result is a consequence of results

(i)	R5243: Finite sum of uncorrelated identically distributed exponential random positive real numbers is a gamma random random positive real number
(ii)	R2506: Independent random real collection is uncorrelated

$\square$