Minimum of Exponential Random Variables has Exponential Distribution

Theorem
Let $\beta_1, \beta_2, \ldots, \beta_n$ be positive real numbers.

Let $X_1, X_2, \ldots, X_n$ be independent random variables.

For each $i$, let $X_i \sim \Exponential {\beta_i}$, where $\Exponential {\beta_i}$ is the exponential distribution with parameter $\beta_i$.

Let:


 * $\displaystyle M = \map {\min_{1 \le i \le n} } {X_i} $

Then:


 * $\displaystyle M \sim \Exponential {\paren {\sum_{i \mathop = 1}^n \frac 1 {\beta_i} }^{-1} }$

Proof
We aim to show that:


 * $\displaystyle \map \Pr {M \le m} = 1 - \map \exp {-m \sum_{i \mathop = 1}^n \frac 1 {\beta_i} }$

for each $m > 0$.

Note that:


 * $\displaystyle M > m$




 * $\displaystyle X_i > m$

for each $i$.

We therefore have:

so:


 * $\displaystyle \map \Pr {M \le m} = 1 - \map \exp {-m \sum_{i \mathop = 1}^n \frac 1 {\beta_i} }$