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Flexible Weibull extension distribution

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

f(x)=(μ+σx2)eμxσ/xexp(eμxσ/x),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)=1exp[eμxσ/x],x>0.F(x) = 1 - \exp[-e^{\mu x - \sigma / x}], \quad x > 0.

respectively, where μ>0\mu > 0, σ>0\sigma > 0 and x>0x > 0.

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

Bebbington, M., C. D. Lai, and R. Zitikis. 2007. “A Flexible Weibull Extension.” Reliability Engineering & System Safety 92 (6): 719–26.