91Ó°ÊÓ

It is given that the random variable X follows exponential distribution with parameter β.

Let X be the random variable follow exponential distribution with parameter β.

i.e. \(X \sim Exp\left( \beta \right)\)

Then, its p.d.f is given by,

\(\begin{array}{l}f\left( x \right) = \left\{ \begin{array}{l}\beta {e^{ - \beta x}};x \ge 0\\0\;\;\;\;\;\;;otherwise\end{array} \right.\;\\\end{array}\)

Therefore, the c.d.f of exponential distribution is

\({F_X}\left( x \right) = \int_{ - \infty }^x {f\left( {{x^'}} \right)d{x^'}} \)

\[ = \int\limits_{ - \infty }^x {\beta {e^{\left( { - \beta {x^'}} \right)}}d{x^'}} \]

\[ = \left[ {\beta \frac{{{e^{ - \left( {\beta {x^'}} \right)}}}}{{ - \beta }}} \right]_{ - \infty }^x\]

\[ = \left[ {{e^{ - \left( {\beta {x^'}} \right)}}} \right]_{ - \infty }^x\]

Therefore,

\[{F_X}\left( x \right) = \left\{ \begin{array}{l}1 - {e^{ - \beta x}}\;;x \ge 0\\0\;\;\;\;\;\;\;\;\;;x < 0\end{array} \right.\]………………………………….(1)

The quantile function is the inverse of the distribution function of \(\alpha \), where \({F^{ - 1}}\left( \alpha \right)\) denotes the \(\alpha \) quantile of x.

i.e. \({Q_x}\left( P \right) = F_x^{ - 1}\left( x \right)\) ………………………………….(2)

The quantile function \[{Q_x}\left( P \right)\] is defined as the smallest \(x\) such that \({F_X}\left( x \right) = 0\).

\({Q_x}\left( P \right) = \min \left\{ {x \in \mathbb{R}|{F_X}\left( x \right) = P} \right\}\)

Thus,

\({Q_x}\left( P \right) = \left\{ \begin{array}{l} - \infty ,\;if\;P = 0\\F_X^{ - 1}\left( x \right),\;if\;P > 0\end{array} \right.\)

Thus, it can be derived by rearranging equation (1),

\(\begin{array}{l}P = 1 - {e^{ - \beta x}}\\{e^{ - \beta x}} = 1 - P\\ - \beta x = \ln \left( {1 - P} \right)\end{array}\)

Therefore,

\(x = - \frac{{\ln \left( {1 - P} \right)}}{\beta }\)

Thus, the quantile function of x is:

\({Q_x}\left( P \right) = \left\{ \begin{array}{l} - \infty ,\;if\;P = 0\\ - \frac{{\ln \left( {1 - P} \right)}}{\beta },\;if\;P > 0\end{array} \right.\)