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The exact values are: mean: mu = a / (a + 1) variance: sigma**2 = a / ((a + 2) * (a + 1) ** 2) skewness: One formula (see https://en.wikipedia.org/wiki/Skewness) is gamma_1 = (E[X**3] - 3*mu*E[X**2] + 2*mu**3) / sigma**3 A short calculation shows that E[X**k] is a / (a + k), so gamma_1 can be implemented as n = a/(a+3) - 3*(a/(a+1))*a/(a+2) + 2*(a/(a+1))**3 d = sqrt(a/((a+2)*(a+1)**2)) ** 3 gamma_1 = n/d Either by simplifying, or by a direct calculation of mu_3 / sigma**3, one gets the more concise formula: gamma_1 = -2.0 * ((a - 1) / (a + 3)) * sqrt((a + 2) / a) kurtosis: (See https://en.wikipedia.org/wiki/Kurtosis) The excess kurtosis is gamma_2 = mu_4 / sigma**4 - 3 A bit of calculus and algebra (sympy helps) shows that mu_4 = 3*a*(3*a**2 - a + 2) / ((a+1)**4 * (a+2) * (a+3) * (a+4)) so gamma_2 = 3*(3*a**2 - a + 2) * (a+2) / (a*(a+3)*(a+4)) - 3 which can be rearranged to gamma_2 = 6 * (a**3 - a**2 - 6*a + 2) / (a*(a+3)*(a+4)) 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J?E?d۳?GTn?8?R1#?kP?g(*T?rrrrrrrr)g D )?g:3P?g9?g?S[?rgOqx?g D )?g9?gɫs ^?gK?gX?g3<֌?g M+?gyS c?gI`s?g辜?g{i?g^0?g?#c?)g%r?gc@v?ghtj}?g(˔^?gHK4U?rğ@_?y}D?焂Rr/?N?YjhG?a0L?4f?<@?@CH?g ސF?g5+-#?g?g?q?r)g}'v?rg*V ?rgr/0+?rg[8 ?rgΥ?rgY?rg#)a?rg?̔?rg]ؚ?rg|c?r)r`rr/ crystalballr7r rrA)rZ calculatedrXr+r+r,test_crystalball_functiones&rcCstgd}tgd}tgd}tjd||}t||ddtgd}tdd tjd d g}||}tjd ||}t||ddttjtjtjd dg}||}tjd||} t|| ddttjtjtjtjdg}||} tjd||} t| | ddttjtjtjtjdg}||} tjd||} t| | ddttjtjtjtjdg}||}tjd||}t||dddS)zg All values are calculated using the pdf formula and the integrate function of Mathematica )rgrrYrgrY)rYrYrgrr)rrrrrrr`rj)gԚh@g [&@gͪ@g2dQ@gn6-@g?ޫV&̿gBgQCRgQF\jrg @gn(@r5g?Җ)r~gMg'#@rg('UHrN)r`r r/rr=rr)r.rZexpected_0th_momentZcalculated_0th_momentrr)Zexpected_1th_momentZcalculated_1th_momentZexpected_2th_momentZcalculated_2th_momentZexpected_3th_momentZcalculated_3th_momentZexpected_4th_momentZcalculated_4th_momentZexpected_5th_momentZcalculated_5th_momentr+r+r,!test_crystalball_function_momentss4rcCsRtdd}|}d\}}}t|||}tt|||}t||dddS)Nr5r~)iri@ Hz>rj) r/rrLr`rrrr7r)cbrrrrr>rr+r+r,test_crystalball_entropys   rcCs\d dd}tgd}tjj|d|d\}}}t|ddd |dksJJt|d dd d S) a? 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