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The cumulant generating function Kx (0) of the random variable X is defined by Kx (0) log E(ex), the logarithm of the moment generating function of X. If the latter is finite in a neigh- bourhood of the origin, then Kx has a convergent Taylor expansion: 1 Kx (0) => kn (X)on n=1 and kn (X) is called the nth cumulant (or semi-invariant) of X. (a) Express k₁ (X), k₂(X), and k3(X) in terms of the moments of X. (b) If X and Y are independent random variables, show that kn (X+Y) = kn(X) + kn (Y). =

The cumulant generating function Kx (0) of the random variable X is defined by Kx (0) log E(ex), the logarithm of the moment generating function of X. If the latter is finite in a neigh- bourhood of the origin, then Kx has a convergent Taylor expansion: 1 Kx (0) => kn (X)on n=1 and kn (X) is called the nth cumulant (or semi-invariant) of X. (a) Express k₁ (X), k₂(X), and k3(X) in terms of the moments of X. (b) If X and Y are independent random variables, show that kn (X+Y) = kn(X) + kn (Y). =

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 17EQ
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The cumulant generating function Kx (0) of the random variable X is defined by Kx (0) log E(ex), the logarithm of the moment generating function of X. If the latter is finite in a neigh- bourhood of the origin, then Kx has a convergent Taylor expansion: 1 Kx (0) => kn (X)on n! n=1 and kn (X) is called the nth cumulant (or semi-invariant) of X. (a) Express k₁ (X), k₂(X), and k3(X) in terms of the moments of X. (b) If X and Y are independent random variables, show that kn (X+Y) = kn(X) + kn (Y). =
Transcribed Image Text:The cumulant generating function Kx (0) of the random variable X is defined by Kx (0) log E(ex), the logarithm of the moment generating function of X. If the latter is finite in a neigh- bourhood of the origin, then Kx has a convergent Taylor expansion: 1 Kx (0) => kn (X)on n! n=1 and kn (X) is called the nth cumulant (or semi-invariant) of X. (a) Express k₁ (X), k₂(X), and k3(X) in terms of the moments of X. (b) If X and Y are independent random variables, show that kn (X+Y) = kn(X) + kn (Y). =
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