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

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Result R4796 on D214: Exponential random positive real number
Subresult of R3776: Probability to win an exponential race

Probability for two independent exponential random positive real numbers to be smaller than the other

Formulation 0

Let

$X \in \text{Exp}(\theta_X)$ and

$Y \in \text{Exp}(\theta_Y)$ each be an D214: Exponential random positive real number such that

(i)	$X, Y$ is an D2713: Independent random collection

Then

$\begin{equation} \mathbb{P}(X \leq Y) = \frac{\theta_X}{\theta_X + \theta_Y} \end{equation}$