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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MGF of "generalised" Negative Binomial distribution

Would the "extended" Negative Binomial have the same MGF as Negative Binomial? (See the definition of "extended" Negative Binomial below by Wikipedia) Could someone please help ...
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How to find the MGF of the max of a set of i.i.d. exponential random variables

As the title suggests, I would like to find the MGF of the max of iid exponential random variables. Assume $Z=\max(x_{1},...,x_{n})$, where $x_{i}$ is distributed as exponential($\beta$) and has pdf $\...
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Proving relation between counts and arrival times via transforms

I'll state what I'm trying to prove below. For a Poisson process $N(t) \sim \operatorname{Poisson}(\lambda t)$, $$ P\left(S_n \leq t\right)=P\left(N(t) \geq n\right)=1-P(N(t)<n), $$ where $S_n=\...
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Chernoff Bounds for Independent Bernoulli Sums

What is wrong with this proof? Can you notice that? or I am wrong? In my opinion, in the R.H.S. of the inequality (3.2), the index of 'e' is negative but it must be positive if we use the given proof ...
Win_odd Dhamnekar's user avatar
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Relationship between CDF and MGF?

Suppose $X$ is a random variable $\in (0,1]$ $f(x)$ is the CDF of this random variable $g(t)=\operatorname{mgf}(-t)$ where "mgf" is the moment generating function. Can we infer the ...
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What distribution has CDF $\propto \psi ^{(1)}\left(x^{-\frac{1}{2}}\right)$

Is there a way to express random variable with the following CDF $g(x)$, $x\ge 0$, in terms of known named distributions? $$g(x)\propto \psi ^{(1)}\left(x^{-\frac{1}{2}}\right)$$ where $\psi^{(k)}$ is ...
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Existence of moment generating function [closed]

Give a example of discrete random variable for which mgf $M(t)=E\left(e^{tx} \right)$ does not exist . I have tried with geometric(p) distribution when $(1βˆ’p)e^tβ‰₯1$ , the mgf does not converge. Is it ...
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Are linear combinations of independent random variables again independent?

Let $X_1,X_2,\ldots,X_n$ be (iid) Random variables and define $Y_n:=\sum_{j=1}^na_jX_j$ with $a_j\in \mathbb{R}$, can we then say that the $a_jX_j$ are independent aswell. Can we express the MGF than ...
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Double Expectation in Moment Generating Function

Context Assume a game using a fair 6 sided dice ends when one rolls a 6. A random variable N is the total number of throws. N is distributed such that: $p_N(n) = \frac{5}{6}^{n-1}\frac{1}{6}$ Assume ...
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Upper Bound of MGF for a non-negative random variable with bounded variance

Let $X$ be a non-negative random variable with finite variance. It is obvious that its MGF $E[e^{-\lambda(X-E[X])}]$ exists for $\lambda > 0$. How to prove that $E[e^{-\lambda(X-E[X])}] \le \exp(\...
Ruiyuan Huang's user avatar
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The distribution function of the min of two random variables which are dependent via a common term

Consider the following two random variables: $$Z_1=U_1-X_1$$ and $$Z_2=U_2-X_1,$$ where $U_1$ and $U_2$ are two i.i.d random variables following a general distribution, and $X_1$ is an exponential ...
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Confirmation of MGF of Shifted Exponential Distribution

Consider the probability distribution $D \sim X - a$ where $X \sim \text{Exp}(\lambda)$. My task is to find the MGF of probability distribution $D$. I think I have a solution but it contradicts what I ...
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Limiting value of the moment generating function

Suppose that the discrete random variable $X_{n}$has a geometric distribution given by $$f_{X_n}(x_n)=P_n{(1-P_n)}^{x_n}$$ where $$x_n={0,1,2,3,}$$ and $$P_n\ =\ \frac{\lambda}{n}$$ for $0<\lambda&...
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Expected value of Y = (1/X) where X is Gamma Distribution [closed]

Let's suppose $$E\left(\frac{1}{x}\right)=\int_{0}^{\infty}{M_X(-t)dt}$$ Could you please help me to find $$E\left(\frac{1}{x}\right)$$ where $$X \sim\textrm{Gamma}(\alpha, \beta)$$ and $$E(X) = \...
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Inequality on the moment generating function of a centered random variable which is bounded above

I am stuck on the first part of problem 8.2 of the book "A Probabilistic Theory of Pattern Recognition" by Luc Devroye: Show that for any $s > 0$, and any random variable $X$ with $\...
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Evaluating $E(x^{-1}). $

Could you please help me to prove the following equation: $$E(x^{-1})=\int_{0}^{\infty}M_{x}(-t)dt$$ Where $M_{x}(-t)$ is the moment-generating function. I think the following equation will be useful: ...
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Calculating the n-th moment of a RV, including negative fractional moments

I am stuck trying to solve the following exercise.. Let $X: \Omega\to [a,b] \subset \mathbb R$ be a uniformly distributed random variable. Compute the n-th moment of $X$, i.e. compute $\mathbb E[X^n]$ ...
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Compound binomial distribution distributed as binomial

Suppose we have independent family of random variables $\{Y\}_{i\in\mathbb{N}}\cup\{N\}$, where $Y$s are identically distributed. Next consider a sum of random number of random variables $W_N\equiv\...
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Moment generating function interval on which is defined

I have a simple question about moment generating functions(MGFs). Does the interval on which a MGF is defined corresponds to the support of the random variable? For instance, considering a standard ...
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Expected value of Bernoulli variable from the moment generating function

I am studying the Moment Generating Functions of discrete random variables and I got an exercise asking to derive the mgf of a Bernoulli variable and its expected value. I start from the definition of ...
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How to find the moment generating function of a random variable X with PDF of 4x^3 for 0<x<1?

Let $X$ be a continuous random variable with PDF \begin{equation} \nonumber f_X(x) = \left\{ \begin{array}{l l} 4x^3 & \quad 0 < x \leq 1\\ 0 & \quad \text{otherwise} \end{array} \right. \...
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Variance of inverse RV [closed]

I am trying to find variance of Y=1/(1+abs(x)) where X is Gaussian RV with mean m, var sigma^2To do that my initial step is to find MGF. can anyone give me ...
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Upper bound for m.g.f

$X$ is a discrete random variable from power series family (e.g., binomial, poisson etc.). is it possible to find an upper bound for the m.g.f of $X$? N.B: from stack exchange I obtained the following ...
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Moment generating function of a constant multiple of a random varaible

Let $Y$ be a random variable which is a function of another random variable $X$, such that $Y=aX$, where $a$ is a constant. Is it possible that the moment generating function (MGF) of $Y$, is given by,...
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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 $M_{V,W} (s_1, s_2)$ of V and W [closed]

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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Derivation of moment generating function for limiting distribution of sum of logbeta distributed variables

A sum of logbeta distributed variables occurs in this question Distribution with a given moment generating function Let, $X_j \sim Beta(j\sigma, 1-\sigma)$, $Y_j = -\log(X_j)$ and $S_n = \sum_{j=1}^n ...
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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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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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What is the expectation of $e^X$, where $X$ is a random variable with a geometric distribution?

if $X$ is a random variable with a geometric distribution how can I calculate $$ E(e^X) $$ I have no idea on how to do that.
JacK'o'Lantern's user avatar
1 vote
1 answer
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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{...
user672518's user avatar
3 votes
0 answers
173 views

Finding the UMVUE of $e^{3\lambda}$ in Poi($\lambda$)

Let $ X = (X_1, ... , X_n)$ iid variables coming from Poisson distribution with mean $\lambda$. Find the UMVUE of $e^{3\lambda}$. I tried understanding the solution below (in the possible duplicate ...
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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 ...
Drunk Deriving's user avatar
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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 ...
E. Kaufman's user avatar
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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'[...
ItsAllPurple's user avatar
1 vote
0 answers
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Proof of poisson distribution as a limiting case of the negative binomial distribution, using the MGF [duplicate]

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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