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combinan
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There are 8 letters in COMBINAN ( A1B3C3I1M3N1O1 )
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COMBINAN - In the mathematical theory of probability, the combinants cn of a random variable X are defined via the combinant-generating function G(t), which is ...
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In the mathematical theory of probability, the Combinants cn of a random variable X are defined via the combinant-generating function G(t), which is defined from the moment generating function M(z) as* * * * * G * * X * * * ( * t * ) * = * * M * * X * * * ( * log * * ( * 1 * + * t * ) * ) * * * {\displaystyle G_{X}(t)=M_{X}(\log(1+t))} * which can be expressed directly in terms of a random variable X as * * * * * * G * * X * * * ( * t * ) * := * E * * [ * * ( * 1 * + * t * * ) * * X * * * * ] * * , * * t * ∈ * * R * * , * * * {\displaystyle G_{X}(t):=E\left[(1+t)^{X}\right],\quad t\in \mathbb {R} ,} * wherever this expectation exists. * The nth combinant can be obtained as the nth derivatives of the logarithm of combinant generating function evaluated at –1 divided by n factorial: * * * * * * c * * n * * * = * * * 1 * * n * ! * * * * * * * ∂ * * n * * * * ∂ * * t * * n * * * * * * log * * ( * G * ( * t * ) * ) * * * * | * * * * t * = * − * 1 * * * * * {\displaystyle c_{n}={\frac {1}{n!}}{\frac {\partial ^{n}}{\partial t^{n}}}\log(G(t)){\bigg |}_{t=-1}} * Important features in common with the cumulants are: * * the combinants share the additivity property of the cumulants; * for infinite divisibility (probability) distributions, both sets of moments are strictly positive. |