Questions tagged [moment-generating-function]

A description of a probability distribution which is related to the Laplace transform. Use also for its logarithm, the cumulant generating function.

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Sum of exponential of MVN dimensions

Consider drawing N random variables $x_{i}$ for $1\leq i \leq N$ from a multivariate normal with $\mu = (\mu_1, .. \mu_i, .. \mu_N)$ and $ \Sigma_{N \times N} = [\sigma_{ij}]$ (equivalently, N ...
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If two distributions have the same moments, how different can they be?

Let us suppose we have two distribution functions $F$ and $G$ with shared domain and also shared moments but not necessarily shared moment-generating functions. I have seen from "Whether ...
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Generating function for Gamma distribution expectation

I encountered this formula in my assignment: $$X\sim \Gamma(\alpha, \beta), 1\le k < \alpha$$ $$ E(X^{-k})=\frac{\beta^k}{\prod^k_{i=1}(\alpha-i)} $$ And I wonder what would happen if $k$ is ...
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What are the moments of the Beckmann distribution?

Let $U=(u_1, u_2)$ and $V=(v_1, v_2)$ be two randomly distributed points on the Euclidean plane assuming bivariate normal distributions $U \sim N(\mu_u, \Sigma_u)$ and $V \sim N(\mu_v, \Sigma_v)$ with ...
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MGF of the product of a exponential and a bernoulli random variable

Let $𝑍=π‘‹π‘Œ$ , where $X$ and $Y$ are independent, 𝑋 ~𝐸π‘₯π‘π‘œπ‘›π‘’π‘›π‘‘π‘–π‘Žπ‘™(0.01) and π‘ŒβˆΌπ΅π‘’π‘Ÿπ‘›π‘œπ‘’π‘™π‘™π‘–(0.3) Is there a way to find the m.g.f of 𝑍? I know that I can find the C.D.F by doing as ...
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Determine the joint moment generating function $M_{V,W} (s1, s2)$ of V and W

So the initial question was: Let X and Y be independent random variables with common moment generating function $m_X(s) = m_Y (s) = e^{s^2/2}$. a) Determine the moment generating function $M_V (s)$ of ...
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Moments of $\text{exp}(-|x|^{1/2})$

I'm supposed to show that all of the moments of the density $\text{exp}(-|x|^{1/2})$ are finite. I'm not convinced this is true though. The $p$th moment is \begin{align*} \mathbb{E}[X^p] &= \int_{-...
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MGF of sample mean of poisson distribution

Let $X_1,X_2,\dots,X_n\stackrel{iid}{\sim}Poiss(\lambda)$ Let mgf of $X_1$ is given by $M_X(t)=e^{\lambda(e^t-1)}$ and let $\bar{X_n}=\frac{1}{n}(X_1+X_2+\dots+X_n)$ Then, by Weak Law of Large Numbers ...
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MGF of the absolute Value of a Skellam RV

I am trying to derive the moment generating function for the absolute value of a Skellam random variable $Skellam(\lambda_1, \lambda_2)$ Suppose $X_1 \sim Pois(\lambda_1)$ and $X_2 \sim Pois(\lambda_2)...
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Finding the probability of sum of independent random variables given their Moment Generating Functions

Let $X_{1}$ and $X_{2}$ be independent random variables with respective moment generating functions as $$M_{X_{1}}(t) = (\frac{3}{4} + \frac{1}{4}e^t)^3 \ , \ M_{X_{2}}(t) = e^{2(e^{t}-1)}$$ , $$...
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How to bound sub-exponential variables?

I am trying to understand bounding sub-exponential variables. Suppose for $t=2,\cdots,n$, we have \begin{equation} u_{t-1}u_t \end{equation} where $u_t$ and $u_{t-1}$ are sub-Gaussian. We know that ...
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Generating function of a random walk

Consider a random walk with $S_n=\sum^n_{i=1}X_i$, where the random i.i.d. steps $X_i$ take values $-1,0,2$ with probabilities $1/9,1/9,7/9$ respectively. Set $S_0=1$. I would like to calculate the ...
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Exam question about compound distributions and MGFs

Studying for a test in course about stochastic processes, here's a test question that I can't fully understand: An insurance company insures its policyholders against damages of a particular kind. ...
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Calculating the $M_X(t)$ of the pdf $f_X(x) = \frac{\sin(x)}{2}$

I am working on calculating the moment generating function for the pdf $f_X(x) = \frac{\sin(x)}{2}$ with the bounds $[0, \pi]$, and here is my attempt although I would like to know whether I have ...
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Calculate moment generating function of normal distribution [duplicate]

The moment generating function of a normal distribution is defined as $M(t) = \int_{-\infty}^\infty e^{tx}\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2} dx$ In a book I’m ...
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Textbook clarification on Taylor expansion of mgf (Casella, Berger)

In Statistical Inference by Casella, Berger (version 2), Section 2.6.1, we have the following expression for the Taylor expansion of the mgf: $$M_X(t)=\sum_{r=0}^\infty \frac{(-1)^r\mu_r't^r}{r!}$$ ...
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Moment Generating function of Multinomial Distribution, very confused

I know somewhat similar questions have been asked but they all seem to have somewhat different answers and I'm kind of confused. I am self learning statistics, and I have the following question that ...
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moment generating function of gamma distribution through log-partition function

How to drive the moment generating function of Gamma distribution using log-partition function? Suppose $X\sim\Gamma(\alpha,\beta)$, gamma distribution with parameter $(\alpha, \beta)$. Then $X$ has ...
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Expected Value using Moment Generating Function

Let $M_Xt$ be the MGF of the random variable $X$. Define $S_X(t)=log_{10}M_xt$. I am trying to show that $\frac{d}{dt}S_X(t)|_{t=0}=E[X]$ Here's what I did: $$E[X]=\frac{d}{dt}M_X(t)|_{t=0}$$ $$\frac{...
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Upper Bound for 2nd Raw Moment of Positive Random Variable

Let $X$ be a random variable with support $(0,\infty)$. All I know about $X$ is the support, finite higher moments, and $\mathbb{E}(X)=\mu$. I am trying to come up with a more tractable upper bound ...
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Moment Generating Function for values $t \neq 0$ [duplicate]

I'm studying statistics using Rice's Mathematical Statistics and Data Analysis. We can find the $r$th moment of a random variable $X$ by taking the moment generating function $M(t)$ of $X$ and taking ...
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Calculus in Moment Generating Function

On page 156 of the Statistics textbook, "Mathematical Statistics and Data Analysis" by John A. Rice, I came up with two questions on the section about Moment Generating Functions: 1. Why ...
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Multivariate Normal Quadratic MGF: Eigendecomposition to Matrix form

If $X \sim \mathcal{N}(\mu, \Sigma)$ is a multivariate normal, then the quadratic $X^TAX$ has moment generating function $$M_{X^TAX}(t)= \frac{1}{\sqrt{\det(I - 2tA\Sigma)}}\exp\left(-\frac{1}{2}\mu'[...
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Proof of poisson distribution as a limiting case of the negative binomial distribution, using the MGF

This question is very similar to one that has previously been answered, but with a different parameterization of the MGF. The question is as follows: Show that the negative binomial distribution, $NB(...
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Help finding MGF of mixture distribution

Im having trouble finding any good resources or examples for finding the conditional distribution of two variables. I've tried using double expectations but cant get it to work out. Thanks I found ...
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Do conditional mgfs relate to the joint mgf similarly to marginal mgfs?

Given a (for example, bivariate) random vector, say $(X,Y)$, with joint distribution $f(x,y)$ and moment-generating function $$M_{XY}(t_1,t_2),$$ we can say that the moment-generating functions of the ...
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tails from cumulant generating function

I am trying to obtain finite-sample tail estimates using the mgf. I found a snippet in a google books preview of a formula giving the survivor function by some kind of inversion of the cumulant ...
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Computing variance from moment generating function of exponential distribution

I'm wondering how to get variance of exp. distribution from the raw variance computed using the moment generating function. Here's my line of reasoning: PDF of Exponential distriution is $$ p_X(x) = \...
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Gamma Distribution satisfying property

How can we prove that gamma random variable $X_{n}$ with parameters $(n,3)$ can satisfy the following relation for some $n$? $$P(X_{n} < n/2) > 0.999$$ I used the definition of density function ...
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Expectation of the product of polynomial & exponential transformations of normal r.v

Let $X \sim \mathcal{N}(\mu, \sigma^2)$. Are there any (1) general formula and (2) references to the general formula for $$ \mathbb{E} (X^n e^{tX}),\; n \in \mathbb{N}, t \in \mathbb{R}$$ in ...
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The log of a Moment Generating Function

I am having trouble understanding the rationale of the following related to the MGF: Function Mx(t)=E[exp(tX)], the expectation exists for all t in a neighbourhood of zero, and X has mean mx, show ...
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How would a moment generating function change if all the random variables are increased by a value

Suppose you have some moment generating function $M_x(t)$ Now all the random variables x are increased by a arbitrary value b. What is the new moment generating value? I tried solving this by moving ...
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Weak convergence of moment generating function

I have the following sequence of rvs $$Z_1 = X_0*Y_0$$ $$Z_{n+1} = Z_n /2 + X_n*Y_n$$ Where $X_n$ and $Y_n$ are independent, with $X_n$ having Bernoulli distribution with p=1/2 and $Y_n$ having ...
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Inverting a moment generating function

If I have a random variable $X$ with pdf $f$ I can compute its MGF as $$ M_X(t) = \int_{-\infty}^\infty e^{tx} f(x)dx. $$ My understanding is that this is basically a Laplace transformation. ...
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Sum of indepedent random variables and a constant

Let $X_1$ and $X_2$ be independent Normal random variables with mean $\mu_1$ and $\mu_2$, and variances $\sigma_1$ and $\sigma_2$. Let $Y = X_2-X_1 + c$, where $c$ is a constant. For notational ...
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1 answer
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How to prove there is no moment generating function for t distribution

I'm struggling to show that the moment generating function for t distribution does not exist. So far I tried to show the moment generating function diverges from its integration but the computation is ...
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How do you calculate the expected value of $E\left\{e^{-|X|}\right\}$ e.g. for Gaussian X?

If $X$ is a random variable, I would like to be able to calculate something like $$E\left\{e^{-|X|}\right\}$$ How can I do this, e.g., for a normally distributed $X$?
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Example of random variable with a unique moment sequence but mgf DNE in a neighborhood of 0

Do you have an example of a random variable $X$ with a unique moment sequence but whose mgf does not exist in a neighborhood of 0? In other words, I'm looking for a counterexample to the converse of ...
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Moment generating function of a conditional distribution

Let $S$ ~ Poisson$(\alpha + \beta)$, and $X|_{S = s}$ ~ Binomial$(s, \alpha/(\alpha + \beta))$, $\alpha > 0, \beta > 0$ Suppose Z = S - X is independent from X. What is the distribution of Z? ...
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n'th cumulant (of a CGF) for exponential family / exponential dispersion model

The n'th cumulant is defined to be the n'th derivative of the CGF (cumulant generating function). $$\kappa_n = \frac{d^n K(t)}{dt^n} |_{t=0} $$ But I'm reading in a book (p.215, chapter5, eq. 5.8) ...
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6 votes
3 answers
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Can we get Moment Generating Function(MGF) from data?

We had couple of good discussions about Moment Generating Function(MGF), here and here. But I still have questions on the applications of it and how can it be useful. Specifically, I can understand ...
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Tail bounds for sample means of i.i.d random variables where the moment generating function exists

I want to figure out the proof of Lemma 4 in the following paper. The lemma states that Let $Y_1, Y_2,\ldots Y_m$ be i.i.d random variables such that $\mathbb{E}\left[e^{zY}\right] < \infty$ ...
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Undefined MGF but all moments finite?

For the lognormal distribution: https://en.wikipedia.org/wiki/Log-normal_distribution The moment generating function is undefined, but all the moments exist and are finite. I thought the moment ...
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How to show that $\frac{X_n-a}{\sqrt n}$ approaches a Gaussian (if $X_n \sim\ \chi_{n-p}^2$)

Given that $X_n \sim\ \chi_{n-p}^2$, I think the moment generating function of $\frac{X_n-a}{\sqrt n}$ is $e^{\frac{-at}{\sqrt n}}(1-\frac{2t}{\sqrt n})^{-\frac{n-p}{2}}$. As specified in problem 4 on ...
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Finding the MGF of a bivariate Normal Distribution [duplicate]

Given ($X$, $Y$) whose MGF is defined as: $$M_{XY}(s, t)=E[e^{sX+tY}]$$ Find $M_{XY}(s, t)$ when $X$ and $Y$ are two jointly normal random variables with $E[X]=\mu_X$, $E[Y]=\mu_Y$, $var(X)=\sigma^...
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Distributions of Quadratic form of a normal random variable

I am looking for ways to prove that the moment generating function of $X'AX$ given that $X \sim N(\vec{\mu}, \vec{\Sigma})$ and $A$ is symmetric is defined as: $$M_{X'AX}(\vec{t})= \frac{1}{|I-2tA\...
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4 votes
2 answers
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Generate Moments of Continuous Uniform Distribution with Moment Generating Functions

I am having trouble generating moments from the moment generating function of the uniform. By the definition of M.G.F, we can calculate: $$ M(t) = \begin{cases} \frac{e^{tb} - e^{ta}}{tb-ta} : t \ne ...
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Moment generating function of non-central Chi-squared distribution with complex mean?

I have random variables $(X_1, \dots, X_k)$ distributed independently according to normal distributions with complex means, i.e. $j\mu_i, i=1\dots k, j^2=-1$, with unit variances. I want to study the ...
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2 votes
1 answer
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How to find the MGF of the difference of 2 random variables

Let $X\sim N(12,4)$ and $Y \sim N(3,1)$ Let $Z = X - Y$ Find the Moment Generating Function of $Z$. I tried finding the expected value of $e$ to the power of $tz$, but this isn't possible to separate ...
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Why is moment generating function (MGF) evaluated at zero?

Although this question touched on MGF exists at neighborhood of 0, I still don't understand why the definition of MGF says $M_X{(t)}$ = E $e^{tX}$ for $t$ in neighborhood of 0. Why must it be 0 but ...
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