Tukey lambda distribution

Article summary

{{Probability distribution | name = Tukey lambda distribution | type = density | pdf_image = | cdf_image = | notation = Tukey( ) | parameters = — shape parameter | support =         if | pdf = \left(\ Q(\ p\ ; \lambda\ ),\ \frac{ 1 }{\ q(\ p\ ; \lambda\ )\ }\ \right) \quad ~\mathsf{ for\ any }~ \quad p \; : \; 0 \leq\ p\ \leq\ 1 | cdf = \Bigl(\ Q(p;\lambda),\ p\ \Bigr) ~~\mathsf{ for\ any }~~ p\; : \; 0 \leq\ p\ \leq\ 1 ~   (general case) \frac{ 1 }{\ e^{-x} + 1\ } \quad ~\mathsf{ if }~ \quad \lambda\ =\ 0 \quad (special case exact solution) | mean = 0 \quad ~\mathsf{ if }~ \quad \lambda > -1\ | median = | mode = | variance = \frac{ 2 }{\ \lambda^2\ } \left(\ \frac{ 1 }{\ 1 + 2\ \lambda\ } - \frac{\ \Gamma\,\!(\lambda + 1)^2\ }{\ \Gamma\,\!(\ 2\ \lambda + 2\ )\ } \right) \quad ~\mathsf{ if }~ \quad \lambda > -\tfrac{\ 1\ }{ 2 }\ \frac{\ \pi^{2}\ }{ 3 } \qquad \qquad ~\mathsf{ if }~ \quad \lambda\ =\ 0 | skewness = 0 \qquad \qquad ~\mathsf{ if }~ \quad \lambda > -\tfrac{\ 1\ }{ 3 }\ | kurtosis = ~ \frac{\ (2\ \lambda + 1)^2 \cdot g_2^2 \cdot \big(\ 3\ g_2^2 - 4\ g_1\ g_3 + g_4\ \big)\ }{\ (\ 8\ \lambda + 2\ ) \cdot g_4 \cdot \big(\ g_1^2 - g_2\ \big)^2}\ -\ 3 \quad \mathsf{ if }~~ \lambda > 0\ , \frac{\ 6\ }{ 5 } \qquad \qquad ~\mathsf{ if }~ \quad \lambda\ =\ 0\ ; \mathsf{ where } \quad g_k\ \equiv\ \Gamma\,\!(\ k\,\lambda + 1\ ) \quad \mathsf{ and }…

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