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:

$\ds M = \map {\min_{1 \mathop \le i \mathop \le n} } {X_i} $

Then:

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

Proof

We aim to show that:

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

for each $m > 0$.

Note that:

$\ds M > m$

if and only if:

$\ds X_i > m$

for each $i$.

We therefore have:

\(\ds \map \Pr {M > m}\)

\(=\)

\(\ds \map \Pr {\bigcap_{i \mathop = 1}^n \set {X_i > m} }\)

\(\ds \)

\(=\)

\(\ds \prod_{i \mathop = 1}^n \map \Pr {X_i > m}\)

Definition of Independent Random Variables

\(\ds \)

\(=\)

\(\ds \prod_{i \mathop = 1}^n \map \exp {-\frac m {\beta_i} }\)

Definition of Exponential Distribution

\(\ds \)

\(=\)

\(\ds \map \exp {-m \sum_{i \mathop = 1}^n \frac 1 {\beta_i} }\)

Exponential of Sum

so:

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

$\blacksquare$