# Questions tagged [moment-generating-functions]

This tag is for questions relating to moment-generating-functions (m.g.f.), which are a way to find moments like the mean$~(μ)~$ and the variance$~(σ^2)~$. Finding an m.g.f. for a discrete random variable involves summation; for continuous random variables, calculus is used.

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### Why finite sub-gaussian norm indicates sub-gaussian distribution?

The sub-gaussian norm of a random variable $\xi$ is defined as follows: $$\|\xi\| = \inf \left\{\lambda > 0 | \mathbb E\left(e^{\xi^2 / \lambda^2}\right) \leq 2\right\}.$$ When the moment ...
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### Moment generating function of $f(x)=1/c$

Moment generating function of $f(x)=1/c$, $0<x<c$. Well, I know the calculation is easy. $M_X(t) = \frac{1}{ct} (e^{tc}-1)$. This question is Exercise 2.30 (a) Statistical Inference Book by ...
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### How do I find the second order moment with a MGF?

Let $X$ be a random variable such that $M_X(t)=e^t M_X(-t)$. Find $E(X)$ and $E\left(X^2\right)$. I know the general procedure that to find the $n$th moment, the $n$th order derivative needs to be ...
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### Derivation of moment generating function of a mixture of random variables

Question Let $Z$, $X$ and $Y$ be random variables, where $Z$ is formed by choosing a sample from $X$ with probability $q$ or a sample from $Y$ with probability $1 - q$. If $X$ and $Y$ are independent ...
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### Is it possible to derive the moment generating function?

Let $X$ be a random variable such that $M_X(t)=e^t M_X(-t)$. Find $E(X)$ and $E\left(X^2\right)$. I know the general procedure that to find the expected value, the first order derivative needs to be ...
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### If random variable $X$ has moments of all orders, does this imply that the moment generating function of $X$ exist?

Suppose $EX^n<\infty$ for any $n$, does this imply that $M_X(t)=Ee^{tX}<\infty$ for all $t$ in some neighborhood of zero, that is, the moment generating function of $X$ exist? If this is not ...
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### Closed form for integer moments of Poisson random variable?

I'm interested in a closed-form (special functions included) for the integer moments of the Poisson distribution. Let $X\sim\operatorname{Poisson}(\lambda)$. Then we have for the moment generating ...
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### Let $\{X_i\}$ are i.i.d.. Given $Y$, how to estimate the probability of $n$ such that $S_n \leq Y < S_{n+1}$, where $S_n = \sum_{i=1}^n X_i$?

Let $\{ X_{i} \}_{i \in \mathbb{N}^{+}}$ are i.i.d. random variables with $\mathbb{E}(X_i) = \mu > 0$ and $\mathbb{E}(X_i^2) = \sigma^{2} > 0$.. Let $S_{k} = X_{1} + X_{2} + \cdots + X_{k}$. ...
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### What happens to skewness as you play the same gamble multiple times?

Suppose there is a probability distribution $p$, over money outcomes. Suppose one draws $N$ times from $p$, with replacement, and takes the average money outcome over these $N$ draws. Call this ...
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### Is a real random variable's moment-generating function the Laplace-Stieltjes transform of its CDF?

Wikipedia gives that the moment-generating function for a real random variable $X$ with cumulative distribution function $F$ equals $M_x(t) = \int \limits_{-\infty}^\infty e^{tx}\, dF(x)$, using the ...
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### Under what conditions is $K^{(2)}(t)$ continuous at $t=0$ where $K(t)$ is the cumulant generating function?

For a random variable $X$ let \begin{align} K(t)= \log (E[e^{tX}]) \end{align} This quantity is known as the cumulant generating function. Question: What is some weak sufficient condition for \begin{...
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### Recurrence relations for even orthogonal polynomials

I have been playing around with the theory of orthogonal polynomials, and it occurred to me that we might be able to build a family of orthogonal polynomials from even powers of a variable $x$. For ...
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### Show that chi-squared random variables are sub-exponential with parameter $4$

Let $X \sim N(0, 1)$ so that $Y = X^2$ is distributed as chi squared with mean $1$. So the moment generating function of $Y$ is: $$\frac{e^{-\lambda}}{\sqrt{1-2\lambda}}$$ $Y$ is subexponential if the ...
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### Unsure about result involving Fibonacci Numbers $\sum_{n = 0}^\infty F_n$

I just starting working through a text on Moment Generating Functions. One elementary example is that of the Fibonacci Sequence and determining it's Moment Generating Function. I've found a result for ...
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