# Questions tagged [moment-generating-functions]

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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### If $X$ is sub-Gaussian random variable with variance proxy $\sigma^2$, how to show that $E\{ \exp( t X^2) \} \leq (1 - 2 t \sigma^2)^{-1/2}$?

If $X$ is sub-Gaussian random variable with variance proxy $\sigma^2$, i.e., $E(X) = 0$ and $E\{ \exp(s X) \} \leq \exp( \frac{\sigma^2 s^2}{2} )$ for $\forall s \in \mathbb{R}$, then how to show that ...
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### Unbiased Cumulant Estimate - Fifth Cumulant

I am searching the definition of the $5^{th}$ unbiased cumulant estimate. Let $K_j$, be the $j$-th unbiased cumulant estimate of a probability distribution, based on the sample moments. Let $m_j$ be ...
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### $\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$, upper bound second moment from first moment

Let $X$ be a non-negative random variable bounded on $[0,1]$. Is it true that $\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$ for some constant $k$? If not, are there any minimal assumptions on $X$ where this ...
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### Quick question about MGFs; Is $M_X(t)$ always strictly positive?

I recently had an assignment where I had to prove that, if $X,Y,Z$ are all independent random variables and $X+Z$ has the same distribution as $Y+Z$ then $X$ has the same distribution as $Y$. These ...
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### Are cumulants the only additive functions of independent random variables?

For a random variable $X$, the cumulant generating function $CGF_X$ is defined as $CGF_X(t)=\log Ee^{tX}$, and the nth cumulant $k_n(X)$ is defined as the coefficient of $t^n/n!$ in the corresponding ...
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### Suppose that $X$ is a random variable with $E(X^{n})=3^{n}(n+1)!$ for $n \geq 1$. Find the distribution of $X$. [closed]

I tried using the moment generating function. I got: $M_{X}(t)=E(e^{tX})=\Sigma_{n=0}^{\infty} (n+1)(3t)^{n}$. But I don't understand which distribution would it be.
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### An inequality involving infimums of the scaled $k$-th moment and of a scaled moment-generating function.

Let $X$ be a non-negative r.v., prove that \inf_{k\in\mathbb{Z}_+} \frac{\mathbb{E}[X^k]}{t^k}\leqslant \inf_{\lambda\geqslant 0}\frac{\mathbb{E}[e^{\lambda X}]}{e^{\lambda t}},\;\...
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### Control of distribution of random variables by moment method

I just learned how finite $p$-th moment control the distribution of random variables. That is, given any random variable $X$ if $E[|X^p|]<\infty$ (or we say $X\in L^p$) for non-negative integer $p$ ...
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### Bound on cumulant generating function of a weighted sum of uniform random variables

Question Define $\mathbf{a} = (a_1, \ldots, a_p)$ where $p$ is a positive integer and the $a_l$ are i.i.d $\text{Uniform}(-1,1)$ random variables. Fix a unit vector $w \in \mathbb{R}^p$. Consider the ...
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### Convergence of MGF of squared norm of sum of iid unit vectors

Suppose I have $N$ iid random vectors $\sigma_1,\ldots,\sigma_N$ that are uniformly distributed in $S^1$. Let $\bar{\sigma}_N:=N^{-1}(\sigma_1+\cdots+\sigma_N)$ denote the sample average. I am ...
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### Moment Generating Function for a specific set of moments - what's the distribution it follows?

While studying a coin-toss experiment inspired by an interview question I stumbled upon a discrete random variable whose moments (after normalizing and centering to zero, that is $\mu=0,\sigma=1$.) ...
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Let $X_1$ be a random variable. I need to prove that the function $f(x_1) = \frac{e^{t_xX_1}}{\varphi(t_x)}$ is a density function, whereby $\varphi(t)$ is the moment generating function of $X_1$ and $... • 876 3 votes 0 answers 90 views ### What did Rota mean by "one can define cumulants relative to any sequence of binomial type"?$\newcommand{\E}{\mathbb{E}}$Near the end of "Finite Operator Calculus" (1976), G.C. Rota writes: Note that one can define cumulants relative to any sequence of binomial type, e.g. the ... • 1,158 2 votes 0 answers 133 views ### Central limit theorem: negative moments Let$X_{1},\ldots,X_{N}$be i.i.d. random variables with mean 0 and variance 1. For simplicity, assume that$X_{1}$has all finite moments. Let S_{N} = \frac{1}{\sqrt{N}}\,\sum_{i=1}^{... • 21 1 vote 0 answers 39 views ### Second moment of a Gaussian distribution here I have a$M\sim \mathbb{N}(0,\sigma^2_N)$and $$M = \int \int_A \frac{\partial}{\partial x} \Bigl(\int^\infty f(x,z,\ell) d \ell\Bigl) dx dz$$ where$f(x,z,\ell)$is also a r.v. I would like ... 1 vote 0 answers 24 views ###$H^1(\mathbb{R})$function has a finite moment? I have a function$u_0(x) \in H^1(\mathbb{R})$. It is nonnegative, compactly supported, and has a mass$M>0$. I have to prove that its first moment is finite. My attempt:$\int_{\mathbb{R}} xu_0(x) ...
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Assume $X$ a continuous non-negative random variable follow a Gamma distribution having shape parameter $k$ and scale parameter $\theta$ with the following CDF \begin{align} F_{X}(x) =\frac{\gamma\...