Flexible Weibull extension distribution

This distribution was proposed by Bebbington (2007). The probability density function \(f(x)\) and cumulative density function \(F(x)\) are given by:

\[f(x) = \left( \mu+ \frac{\sigma}{x^2} \right) e^{\mu x - \sigma / x} \exp \left( -e^{\mu x - \sigma / x} \right),\]

and

\[F(x) = 1 - \exp[-e^{\mu x - \sigma / x}], \quad x > 0.\]

respectively, where \(\mu > 0\), \(\sigma > 0\) and \(x > 0\).

Next figure shows possible shapes of the \(f(x)\) and \(F(x)\) for several values of the parameters.