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 ...
Zifeng Zhang's user avatar
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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 ...
claudioclaudio's user avatar
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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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2 answers
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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 ...
bears's user avatar
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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.
Tas's user avatar
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Is $(e^{i \lambda B_t + \frac{1}{2}\lambda^2t})_{t\geq 0}$ a martingale?

Showing that $(e^{\lambda B_t - \frac{1}{2}\lambda^2t})_{t\geq 0}$ is a $\mathbb{R}$-valued martingale Let $B$ be a standard $\mathbb{R}$-valued Brownian motion and let $\lambda\in\mathbb{R}$. From $...
Wilfred Montoya's user avatar
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Moment generating function of squared norm of multivariate Gaussian

Let $X \in \mathbb{R}^d$ be a zero-mean multivariate Gaussian, with independent components, i.e., $X \sim \mathcal{N}(0,\Sigma)$, with $\Sigma = diag(\sigma_1^2,\ldots,\sigma_d^2)$ a diagonal matrix. ...
funny_name's user avatar
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Moment Generating Functions and Probability

Let $X_n$ be the size of the $n$-th generation of a branching process, with family size probability generating function $G(s)$. Let $X_0 = 1$. Suppose the family-mass size mass function is $P(X_1 = k) ...
Timothy Ho's user avatar
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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 \begin{equation} \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}},\;\...
o.spectrum's user avatar
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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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A question involving the expectation of the random variable $2^X$

Let $X$ be a random variable with mgf $M_X(t)=a+be^{2t}$.Given that mean of the random variable $X$ is $1.5$. Then find the expected value of the random variable $2^X.$ My attempt: We know that $E(X^r)...
MathRookie2204's user avatar
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1 answer
31 views

MGF dominated by an exponential function

Let $X$ be a mean-zero random variable whose moment generating function is bounded by a symmetric exponential function, e.g: $$ \mathbb{E}[e^{\lambda X}] \leq \exp(c|\lambda|) $$ with $c > 0$. ...
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Does the moment generating function exist?

Suppose $\nu$ is a probability measure on the positive integers $\{1,2,3\dots,\}$, and let $\pi_{a}$ and $\pi_b$ be Poisson distributions on the positive integers with parameter $a,b$ respectively, ...
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How is it possible for the mean of a log-normal distribution to have units?

I have a set of measurements $a$ (units m) which are log-normally distributed (with parameters $\mu$ and $\sigma$). The expected value (or mean) of $a$ is just the first arithmetic moment, i.e. $$ E(a)...
Plagioclase's user avatar
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2 answers
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Prove that for all $k \in \mathbb{N}^*$, $\mathbb{P}(S \ge k) \le \dfrac{1}{2^n}\left(\dfrac{en}{k}\right)^k$

Problem: Let $b$ be a random variable such that $\mathbb{P}(b=0) = \mathbb{P}(b=1) = \dfrac{1}{2}$. Let $b_1,\ldots,b_n$ be independent copies of $b$ and $S = \sum_{i=1}^n b_i$. Prove that $$\forall k ...
Tung Nguyen's user avatar
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gamma distribution is to exponential distribution what ---- distribution is to Laplace distribution?

Context I'm working on a random walk problem with an two-sided exponential distribution (i.e., a Laplace distribution). From [1], I know that "The sum of $n$ independent $\operatorname{Exp}(\...
Michael Levy's user avatar
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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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Computing expectation of $\frac{X}{X+Y}$ using moment generating functions only

I have two positive, independent, absolutely continuous random variables $X$ and $Y$ with their moment generating functions $M_X$ and $M_Y$. In Foata Fuchs, there is an exercise stating $$\mathbb E\...
Peter Strouvelle's user avatar
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1 answer
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$X_1,....,X_n$ i.i.d. random variables $\sim \mathcal{N}(\mu, \sigma^2 )$, $Y = (X_1-\bar{X}, ..., X_n -\bar{X})$, find MGF of Y.

The question: Let $X_1,....,X_n$ i.i.d. random variables $\sim N(\mu, \sigma^2 )$, and $Y = (X_1-\bar{X}, ..., X_n -\bar{X})$, show that $$M_Y(t) = \exp \left\{ \frac{\sigma^2}{2} \sum_{i=1}^n (t_i - \...
The Math Hermit's user avatar
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0 answers
90 views

Joint cumulants of repeated random variables

I'm trying to prove that: $$\kappa_{k_1,\dots,k_n}(X_1,\dots,X_n)=\kappa_{\underbrace{1,\dots,1}_{k_1+\dots+k_n}}(\underbrace{X_1,\dots,X_1}_{k_1},\dots,\underbrace{X_n,\dots,X_n}_{k_n})$$ where $X_1,\...
Leonardo's user avatar
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1 answer
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Generate three random variable to have specific coskewness

I want to generate three random variable X, Y, and Z that have a specific coskewness. I am defining coskewness between three random variable instead of between two random variable which seems to be ...
Ashkan's user avatar
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Upper bound for $\min_{x \geq 0} e^{t x} \cdot \int_0^\infty e^{-x u} f(u) du$

I am dealing with the following function: $$m(t) = \min_{x \geq 0} e^{t x} \cdot \int_0^\infty e^{-x u} f(u) du,$$ where $t \geq 0$ and $f(u)$ is a pdf of a random variable $X$ distributed over $[0, \...
mike's user avatar
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2 answers
112 views

Prove $\frac{e^{t_xX_1}}{\varphi(t_x)}$ is a probability density function

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 $...
user996159's user avatar
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 ...
Daigaku no Baku's user avatar
2 votes
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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 \begin{equation} S_{N} = \frac{1}{\sqrt{N}}\,\sum_{i=1}^{...
Ele's user avatar
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1 vote
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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 ...
bowellarge's user avatar
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$H^1(\mathbb{R})$ 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) ...
ali's user avatar
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0 answers
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Transformation of Random variables on MGF

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\...
Math Explorer's user avatar
1 vote
1 answer
41 views

Distributions convergence and standard normal random variable using characteristic functions

Suppose $\lambda_{1}$, $\lambda_{2},\ldots\lambda_{n}\ldots$ be a monotone sequence of positive real numbers that go to infinity. Let $X_{n}$ be sequence of random variables with $\Gamma(\lambda_{n},...
maths and chess's user avatar
2 votes
0 answers
81 views

Finding PDF for Irwin-Hall distribution given the MGF [Update]

The problem is the following: Given $X$ a random variable with MGF defined by $$\psi_X(t)=\left(\frac{e^t-1}{t}\right)^n$$ deduce that the pdf of $X$ is $$f_X(x)=\sum_{k=0}^n(-1)^k\binom{n}{k}\mathbf{...
Luigi Traino's user avatar
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1 answer
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Taking the second derivative of the moment generating function.

I am following along from a proof of the central limit theorem using moment generating function from this lecture. At one stage, we are taking derivatives of the moment generating function of an r.v. $...
Joseph's user avatar
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Finding a probability given a mgf

A viral meme starts in one account, and is re-shared by M other accounts, where M is a non-negative discrete random variable. Assume that the future behaviour following from each of the initial re-...
liam song's user avatar
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Finding a probability from a mgf

I am given the question: A viral meme starts in one account, and is re-shared by $\mathbf{M}$ other accounts, where $\mathbf{M}$ is a non-negative discrete random variable. Assume that the future ...
liam song's user avatar
1 vote
1 answer
68 views

Moment generating function of the sum of uncorrelated random variables

Suppose that $\theta_1,\theta_2,\theta_3 \sim\mathcal{U}_{(0,2\pi)}$ are i.i.d. random variables uniformly distributed on $(0,2\pi)$. We define the random variables: $c_{12} = \cos(\theta_1-\theta_2)$...
MathRevenge's user avatar
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1 answer
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Probability distribution from moment-generating function? [closed]

The moment-generating function of $X\sim\chi^2(\nu)$ is given by $$M_X(t)=(1-2t)^{-\nu/2}$$ for all $t<1/2$. For $\chi^2$ distribution, the degree of freedom $\nu$ is positive. Is it possible that ...
ashpool's user avatar
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2 votes
2 answers
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Finding the mgf, expectation and variance of random sum of Poisson random variables

Question Consider the random sum $$Y = I(N>0)\sum_{n=1}^{N}X_n$$ where $\left(X_n \right)_{n\geq 1}$ is a sequence of independently and identically distributed random variables that is independent ...
Hmmmmm's user avatar
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3 votes
2 answers
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Convergence of exponential of Gaussian random walk

Let $\xi_1, \xi_2, \ldots$ be iid. standard normal random variables. (Recall their moment generating function: $\mathbb{E}\left(\mathrm{e}^{\lambda \xi_i}\right)=\mathrm{e}^{\lambda^2 / 2}$.) Let $a, ...
veganwithabeef's user avatar
1 vote
1 answer
176 views

Sub-gaussian norm vs. variance proxy

When studying sub-gaussian variables, I have come across several definitions. One of the most common uses the concept of a "variance proxy," i.e. a (mean-zero) random variable $X$ is sub-...
LSK21's user avatar
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1 answer
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How to find the generating function of a compound random variable?

I have the following compound binomial random variable: $B_n \sim \operatorname{Binomial}(X_n, 1-p)$, where $X_n$ is itself another random variable. This means that $(B_n \mid X_n = x_n) \sim\...
Shatarupa18's user avatar
1 vote
1 answer
74 views

Inverting a moment generating function with simulation

I need to solve the following equation for $\lambda$ involving the moment generating function of a positive random variable, $T$: $$E_T[\exp(-\lambda T)] = q$$ Here, $0<q<1$ and I am able to ...
Rohit Pandey's user avatar
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calculate joint MGF

Let $X_1$ and $X_2$ be independent standard normal random variables. Let $Y_1=X_1+X_2$ and $Y_2=X_1^2+X_2^2$. Show that the joint moment generating function of $Y_1$ and $Y_2$ is $\frac{1}{1-2 t_2} \...
user1168440's user avatar
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0 answers
21 views

What is the density of log-normal geometric Brownian motion $x=log\left(\frac{S}{K}\right)$

If $S_t$ is a GBM with $$S_t \sim LN\left(S_0 e^{\mu t + \frac{1}{2} \sigma^2 t} , \quad S_0e^{2 \mu t + \frac{1}{2} \sigma^2 t}\left(e^{\sigma^2 t} -1\right)\right) $$ Then if $x=log\left(\frac{S}{K}...
THAT'S MY QUANT MY QUANTITATIV's user avatar
4 votes
2 answers
318 views

Formulas or approximations for $\mathbb{E}\left( \frac{X}{\|X\|} \right)$ or $\mathbb{E}\left( \frac{X}{\|X\|^2} \right) $, $X\sim N(\mu, Id)$?

I want to compute or approximate the following expected values with some analytic expression: $\mathbb{E}\left( \frac{X}{\|X\|} \right)$ and $\mathbb{E}\left( \frac{X}{\|X\|^2} \right) $, where $X \in ...
dherrera's user avatar
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How to verify the relationship between the generating function and the moment generating function?

I calculated the Moment Generating Function of a Poisson random variable: $N \sim \mathrm{Poiss}(\lambda)$ $$ E[e^{t N}] = \sum\limits_{k=0}^\infty e^{t k} \frac{e^{-\lambda}\lambda^k }{k!} = e^ {\...
Shatarupa18's user avatar
1 vote
1 answer
57 views

Concentration inequality for $\|A\|_{op}$

Let $A \in \mathbb R^{n\times m}$ be a matrix with independent $1-$subgaussian entries. I have the following bound on the expectation which I derived using a discretization bound: $$\mathbb E \|A\|_{...
dmh's user avatar
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About the conditions of a theorem involving moment generating functions

According to the book "Mathematical Statistics" written by Wiebe R. Pestman, we have the following theorem: Let $X$ be a random variable and $(X_n)_{n\in\mathbb{N}}$ a sequence of random ...
rfloc's user avatar
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Upper bound for a squared zero-mean sub-Gaussian random variable: Transforming a sub-Gaussian rv to a a Gaussian rv

I'm studying from the book "Mathematical Analysis of Machine Learning Algorithms" by Tong Zhang. Theorem 2.9 states Let $\{X_n\}_{n=1}^N$ be independent zero-mean sub-Gaussian random ...
Gerardo Duran-Martin's user avatar

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