Question

Let $X$ be a random variable. If $R(t)=E(e^{t(X-b)})$ for $-h<t<h$, show that if $m$ is a positive integer, then $R^{(m)}(0)=E[(X-b)^{m}]$.

score 80 · Answer 1 · 2021-11-17 01:39:46

Let $X$ be a random variable. If $R(t)=E(e^{t(X-b)})$ for $-h<t<h$, show that if $m$ is a positive integer, then $R^{(m)}(0)=E[(X-b)^{m}]$.

score 51 · Answer 2 · 2021-11-19 11:28:52

51

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List Question

Let $X$ be a random variable. If $R(t)=E(e^{t(X-b)})$ for $-h<t<h$, show that if $m$ is a positive integer, then $R^{(m)}(0)=E[(X-b)^{m}]$.

How to find the $M(0)$ for moment generating function? ($E(X^n) = (-1)^n\frac{1}{5} + \frac{2}{5}$)

Statistical inference, MGF of sum of i.i.d. random variables

Is the moment-generating-function always continuous?

How can you use the moment generating function of a random variable to prove that the probability of its sample space is 1?

Property of zero mean and unit variance random variable

MGF of Random Variables

How to find an expression for an MGF

Joint distribution of diagonal of Wishart matrix

Limit of Moment Generating Function

Moment Generating Function of a Truncated Gaussian

Moment Generating Function of a Truncated Sub-Gaussian

A Conjecture on the Moment Generating Function of Functions of Sub-Gaussian Random Variables

Inequality on the moment generating function of a centered random variable which is bounded above

Finding independent Gaussian variables from moment generating functions

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