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Percentile is calculated by the ratio of the number of values below 'x' to the total number of values. The Percentile Formula is given as, Percentile (P)=(Number of Values Below 鈥渪鈥� Total Number of Values)脳100.

The exponential distribution of \(X\)with the parameter \(\lambda \) is assumed. The exponential distribution cdf can therefore be represented as

\(F(X;\lambda ) = \left\{ {\begin{array}{*{20}{l}}0&{y < 0}\\{1 - {e^{ - \lambda y}}}&{y \ge 0}\end{array}} \right.\)

By \({\eta _p}\), we can designate the \(100{p^{th}}\)percentile. Then we are aware that

\(F\left( {{\eta _p}} \right) = p\)

\(1 - {e^{ - \lambda {\eta _p}}} = p\)

\({e^{ - \lambda {\eta _p}}} = 1 - p\)

\({\eta _p} = - \frac{{\ln (1 - p)}}{\lambda }\)

We know that the meadin \((\tilde \mu )\) percentile of the distribution is \({50^{{\rm{th }}}}\).

\(\tilde \mu = - \frac{{\ln (1 - 0.5)}}{\lambda }\)

\(\tilde \mu = \frac{{0.693}}{\lambda }\)

Definition: Let \(p\)be a number between 0 and 1. The \({(100p)^{th}}\)percentile of the distribution of a continuous rv \(X\), denoted by \({\eta _p}\)), is defined by

\(p = F\left( {{\eta _p}} \right) = \int_{ - \infty }^{{\eta _p}} f (y)dy\)

For median: \({\bf{p}} = 0.5\)

Therefore, the general expression is \({\eta _p} = - \frac{{\ln (1 - p)}}{\lambda }\;\;\;and\;\;\;\tilde \mu = \frac{{0.693}}{\lambda }\).