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

Result R5276 on D214: Exponential random positive real number
Subresult of R3776: Probability to win an exponential race

Probability to win an I.I.D. exponential race

Formulation 0

Let $T_1, \ldots, T_N \in \text{Exp}(\theta)$ each be an D214: Exponential random positive real number such that

(i)	$T_1, \ldots, T_N$ is an D2713: Independent random collection

Let $k \in \{ 1, \ldots, N \}$ be a D5094: Positive integer.

Then \begin{equation} \mathbb{P}(T_k = \min(T_1, \dots, T_N)) = \frac{1}{N} \end{equation}

Proofs

Proof 0

Let $T_1, \ldots, T_N \in \text{Exp}(\theta)$ each be an D214: Exponential random positive real number such that

(i)	$T_1, \ldots, T_N$ is an D2713: Independent random collection

Let $k \in \{ 1, \ldots, N \}$ be a D5094: Positive integer.

This result is a particular case of R3776: Probability to win an exponential race. $\square$