The cumulant generating function is K(t) = log(p / (1 + (p − 1)e t)). The first cumulants are κ 1 = K′ (0) = p −1 − 1 , and κ 2 = K′′ (0) = κ 1 p −1 . Substituting p = ( μ + 1) −1 gives K ( t ) = −log(1 + μ (1−e t )) and κ 1 = μ . See more In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose … See more • The constant random variables X = μ. The cumulant generating function is K(t) = μt. The first cumulant is κ1 = K '(0) = μ and the other cumulants … See more • For the normal distribution with expected value μ and variance σ , the cumulant generating function is K(t) = μt + σ t /2. The first and second derivatives of the cumulant generating function are K '(t) = μ + σ ·t and K"(t) = σ . The cumulants are κ1 = μ, κ2 = σ , and κ3 … See more A negative result Given the results for the cumulants of the normal distribution, it might be hoped to find families of … See more The cumulants of a random variable X are defined using the cumulant-generating function K(t), which is the natural logarithm of the See more The $${\textstyle n}$$-th cumulant $${\textstyle \kappa _{n}(X)}$$ of (the distribution of) a random variable $${\textstyle X}$$ enjoys the following properties: • If $${\textstyle n>1}$$ and $${\textstyle c}$$ is … See more The cumulant generating function K(t), if it exists, is infinitely differentiable and convex, and passes through the origin. Its first derivative ranges monotonically in the open interval from the infimum to the supremum of the support of the probability distribution, and its … See more Weband the cumulant generating function is the sum K S ( ξ ) = K X ( ξ )+ K Y ( ξ ) . Consequently, the r th cumulant of the sum is the sum of the r th cumulants.
TOPIC. Cumulants. Just as the generating function M tions …
WebNov 13, 2024 · 在上式中， z 可以被视为natural parameter，cumulant generating function则为： \varphi(z) = log\frac{f(z)}{\frac{1}{\sqrt{2\pi}}exp(-\frac{z^2}{2})} ，对其 … WebIn probability, a characteristic function Pˆ( k) is also often referred to as a “momentgenerating function”, because it conveniently encodes the moments in its … consult with stakeholders
Cumulants - Scholarpedia
http://www.scholarpedia.org/article/Cumulants WebFeb 10, 2024 · The k th-derivative of the cumulant generating function evaluated at zero is the k th cumulant of X. Title: cumulant generating function: Canonical name: CumulantGeneratingFunction: Date of creation: 2013-03-22 16:16:24: Last modified on: 2013-03-22 16:16:24: Owner: Andrea Ambrosio (7332) Last modified by: Andrea … edward g robinson in the ten commandments