Editing Exponential distribution (section)

===Distribution of the minimum of exponential random variables===
Let ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> be [[Independent random variables|independent]] exponentially distributed random variables with rate parameters ''λ''<sub>1</sub>, ..., ''λ<sub>n</sub>''.  Then
<math display="block">\min\left\{X_1, \dotsc, X_n \right\}</math>
is also exponentially distributed, with parameter
<math display="block">\lambda = \lambda_1 + \dotsb + \lambda_n.</math>

This can be seen by considering the [[complementary cumulative distribution function]]:
<math display="block">\begin{align}
      &\Pr\left(\min\{X_1, \dotsc, X_n\} > x\right) \\
  ={} &\Pr\left(X_1 > x, \dotsc, X_n > x\right) \\
  ={} &\prod_{i=1}^n \Pr\left(X_i > x\right) \\
  ={} &\prod_{i=1}^n \exp\left(-x\lambda_i\right) = \exp\left(-x\sum_{i=1}^n \lambda_i\right).
\end{align}</math>

The index of the variable which achieves the minimum is distributed according to the categorical distribution
<math display="block">\Pr\left(X_k = \min\{X_1, \dotsc, X_n\}\right) = \frac{\lambda_k}{\lambda_1 + \dotsb + \lambda_n}.</math>

A proof can be seen by letting <math>I = \operatorname{argmin}_{i \in \{1, \dotsb, n\}}\{X_1, \dotsc, X_n\}</math>. Then,
<math display="block">\begin{align}
\Pr (I = k) &= \int_{0}^{\infty} \Pr(X_k = x) \Pr(\forall_{i\neq k}X_{i} > x ) \,dx \\
      &= \int_{0}^{\infty} \lambda_k e^{- \lambda_k x} \left(\prod_{i=1, i\neq k}^{n} e^{- \lambda_i x}\right) dx \\
      &= \lambda_k \int_{0}^{\infty}  e^{- \left(\lambda_1 + \dotsb  +\lambda_n\right) x}  dx \\
      &= \frac{\lambda_k}{\lambda_1 + \dotsb + \lambda_n}.
\end{align}</math>

Note that
<math display="block">\max\{X_1, \dotsc, X_n\}</math>
is not exponentially distributed, if ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> do not all have parameter 0.<ref>{{cite web|last1=Michael|first1=Lugo|title=The expectation of the maximum of exponentials| url=http://www.stat.berkeley.edu/~mlugo/stat134-f11/exponential-maximum.pdf|access-date=13 December 2016|archive-url=https://web.archive.org/web/20161220132822/https://www.stat.berkeley.edu/~mlugo/stat134-f11/exponential-maximum.pdf |archive-date=20 December 2016|url-status=dead}}</ref>