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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17 views

How to calculate the the number of nonzero for Poisson binomial distribution

Suppose a sequence of random number $S$, each entry has probability $p_i$ to be 1, otherwise choose zero. The sequence of probability is not necessary identical. According to wiki, we can compute the ...
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28 views

condition on the moment generating function

I have two random variables $X$ and $Y=-X$. What condition must holds on the moment generating function such that $Y$ and $X$ have the same distribution? I know that if $\psi_X(t) = \psi_Y(t)$ then $X$...
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Domain of Moment Generating Function Bernoulli

I want to find the domain of a Moment Generating Function of random variable k following a Bernoulli distribution. This means that: \begin{equation} f(k;p) = \left\{ \begin{array}{cc} ...
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30 views

How do I calculate the mean and variance of a uniform distribution using the moment generating function?

Random variable X with uniform distribution over the interval (2,7) Find the moment generating function and use it to obtain the mean and variance of X. I was able to find the mgf which is given by: (...
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If joint distribution of $X, Y$ is $e^{-y}I_{0<x<y<\infty}$, find the mgf of $Y-X$

Let joint distribution of $X, Y$ be $e^{-y}I_{0<x<y<\infty}$. Find the mgf of $Y-X$. Mgf of $x$ is $1/(1-t_1)$ and mgf of $y$ is $1/(1-t_2)^2$. I know that $x$ and $y$ are not independent, so ...
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52 views

Defining a conditional moment generating function

Let $X$ be a bernoulli random variable with parameter $p\in (0,1)$. Then the moment generating function is given by $M_X(t)=1-p+pe^t$. Now suppose $(Y\mid Z=z)\sim Ber(z)$ where for example $Z$ is ...
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Can we conclude that the given stochastic process $X_t \sim \mathcal{N}(\mu t,\sigma^2 t)$?

We know that for any $\beta \in \mathbb{R}$, the following equality holds: $$ \mathbb{E}\left[\exp(\beta X_t)\right]=\exp\left(\mu \beta t+\frac{1}{2} \sigma^2 \beta^2 t\right) $$ Can we conclude that ...
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PDF derivation, inverse Laplace of $\exp \left\{-\mathbf{m}^T ( \mathbf{I}s^{-1} + \mathbf{C})^{-1} \mathbf{m}\right\}$

StackExchange community! I am trying to derivate a probability density function (PDF) (say $f(x)$). I have derived the moment generating function (MGF) given by $$\mathcal{M}(s)= \mathcal{A}\exp \...
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Can cumulants arise from another property of the distribution.?

Recall that cumulants are defined as \begin{align} k_n = \frac{d^n}{d t^n}\log (M_X(t)) |_{t=0}, n \in \mathbb{N}, \end{align} where $M_X(t)$ is the moment generating function of a random variable $X$....
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Taylor series expansion in moment-generating function

I am working through an example in Wackerly's Mathematical Statistics and it asks to find the moment-generating function $m(t)$ for a Poisson distributed random variable with mean $\lambda$. $$m(t)=E(...
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$\chi^2$ function problem - moment generating functions

Consider $X$ to be such that $X$ is distributed $\chi_m^2$ for a natural number $m$. I want to do the following problems: (i) Prove that the MGF (Moment Generating Function) of $X$ is $m_X(t)=\frac{1}{...
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Meaning of parameter t in moment generating function.

Moment generating function for a random variable X is given by M(t)=$E[e^{tX}]$ for some −h < t < h. Why does −h < t < h ?. I am not able to understand the importance and meaning of ...
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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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Moment generating function of a discrete random variable that takes 0.5 or -0.5 with equal probabilities

Having a bit of a disagreement with a textbook I'm working through... the following question is from Schaum's Outlines of Probability and Statistics, 4th. edition. Problem 3.63. The question is as ...
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Why are odd moments of Normal distribution equal to 0

I am studying the topic of MGF and have come across the result that odd moments of $N(0,1)$ are $0,$ yet I cannot see why. Given $\mu_k$ the $k$-th moment and from the fact that: \begin{equation} M(t)=...
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46 views

Prove the Wald's identity using that $E[S_{N}]=M^{\prime}_{S_{N}}(0)$

Let $X_{1}, X_{2}, \ldots$ i.i.d., and $N$ be an independent variable with positive integer values. Let $S_{N}=$ $X_{1}+\cdots+X_{N}$ be the random sum. Let us assume that the moment-generating ...
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Can it ever be that for a random sample $X_1, ..., X_n$ we have that $\frac{1}{n}\Sigma_{i=1}^n x_i^2 \lt (\frac{1}{n}\Sigma_{i=1}^n x_i)^2$

I have had a homework problem about using method of moments for estimating a uniform random variable. I probably made a calculation mistake because as was pointed out to me, we've shown in some ...
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Laplace transform of the Weibull distribution for $\kappa>1$

In the Wikipedia article on the Weibull distribution, the moment generating function is given: $$g(s) = \sum\limits_{n=0}^\infty \frac{s^n \lambda^n}{n!} \Gamma\left(1+\frac{n}{\kappa}\right )$$ ...
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skewness and kurtosis of the sum and difference of correlated random variables

In the same way that the variance of the sum (or difference) of correlated variables can be determined (e.g. see here - https://en.wikipedia.org/wiki/Variance#Sum_of_correlated_variables) is there an ...
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Show that $M\left(t\right)$ is the expected number of values ​in the sample that lie between $\mu-t$ and $\mu+t$

Let $X_1,...,X_{100}$ be a 'mathematical sample' from the $N(\mu, \sigma)$ distribution, with other words, they are independent and identically distributed random variables. Let $$M\left(t\right)=\...
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Solving for $s$ in $E[(1-h)^s]$ in terms of moments of $h$

Suppose $h$ is a discrete random variable with probability of seeing $h_i$ proportional to $h_i$, and moments $M_k=E[h^k]$ exist for $k=1,2,\ldots$. Is it possible to get bounds on $s$ where the ...
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Where are the imaginary components in a moment generating function (MGF) of a distribution?

An MGF is $$M_X(t)=E(e^{tX})=\int_{-\infty}^\infty e^{tx}f_X(x) dx$$ whereas a Laplace transform is $$\mathcal L\{f_X(x)\}(s)= \int_0^\infty e^{-sx}f_X(x)dx$$ I am not referring to the sign difference,...
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Calculating the moment generating function: $t^k(1-p)^{k-1}p$

I was calculating the moment generating function, then got stuck and looked at the solutions at the end of the book. They show that: $M_X(t) = E_X(t^X)=\sum_{k=1}^{\infty} t^k(1-p)^{k-1} = \frac{p}{1-...
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Moment generating functions of $(1-X)/4$ where $X\sim U[0,1]$

Approach A. Note that $Z=(1-X)/4\sim U[0,1/4]$, hence by using the moment generating function for (continues) uniform distribution we have that $$M_Z(t)=\frac{4(e^{t/4}-1)}{t}.$$ Approach B. $$M_Z(t)=\...
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What is $E\left[\left(X_{i}-\overline{X}\right)^{3}\right]$? [closed]

We all know $E\left[\left(X_{i}-\overline{X}\right)^{2}\right] = \frac{n-1}{n}\sigma^{2} = \frac{n-1}{n}E\left[\left(X_{i}-\mu\right)^{2}\right]$ where $\overline{X}$ is the average of a sample with ...
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Proving $e^{tx}$ is the only possibility for moment generating function

I'm interested in showing that the choice of the expression for moment generating functions $M_X(t)=E(e^{tx})$ for probability distributions is entirely fixed, in that there is no other general ...
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53 views

What is the moment generating function of this random variable?

we are given $n$ independent standard Gaussian random variables $x_i \sim N(0,1)$ and deterministic integer coefficients $c_i\geq 1$. What is the moment generating function of $Z=(\sum_{i=1}^n c_ix_i)^...
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Let $Z\sim N(0,a)$ with variance $a>0$. Show that the mgf of $Z^{2}$ is $M_Z^{2}(t)=\frac{1}{\sqrt{1-2at}}$

Let $Z\sim N(0,a)$ with variance $a>0$. Show that the mgf of $Z^{2}$ is $M_Z^{2}(t)=\frac{1}{\sqrt{1-2at}}$ My Working: Now using the density function of normal random variable with mean $0$ ...
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89 views

Smoothness of the logarithm of moment generating functions

This is Problem $2.6$ from the text on concentration inequalities by Boucheron, Lugosi and Massart. Let $Z$ be a real-valued random variable. Show that the set $$S = \{\lambda \geqslant 0 \mid \...
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40 views

How does the variance appear in $\frac{E(X-\mu)^3}{\sigma^3}$

I wanted to know how the variance appears in the following moment about the mean when calculating for skewness: $\frac{E(X-\mu)^3}{\sigma^3}$ By expanding it we get: $$\frac{E(X^3)-3\mu[E(X^2)+\mu E(X)...
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Moment generating function problem with integral

I have a random variable $X$ with density function \begin{equation} f(x)=Nx^{-\alpha}e^{-x}\mathbb{1}_{[1,\infty)}(x) \end{equation} To continue with the main part of the exercise I am working on I ...
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LOTUS for Laplace Space

Let $f(t)$ be the density of a non-negative, absolutely continuous random variable $T$. While $f(t)$ has no closed form representation, suppose its Laplace transform $$ \int_0^\infty e^{-st}f(t)\,\...
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61 views

Probability of the sum of dice to be a certain number

Let's roll $5$ white dice and $5$ red dice and calculate the number $S = \text{(sum of the while dice)} - \text{(sum of the red dice)}$. What is the probability for this sum to be $0$ and what is the ...
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Find the variance of $S_N$ in terms of the means and variances of $X$ and $N$

I got this problem from Introduction to Probability by Anderson D, Seppalainen T, Valko B. It is challenge problem 10.55. Let $X_1,X_2,...$ be i.i.d. random variables and $N$ an independent ...
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Variance of Pearson's chi-squared statistic

Let $\nu=(\nu_1,\ldots,\nu_r)$, $\sum_j \nu_j=n$, be multinomially distributed with parameter vector $p=(p_1,\ldots,p_r)$ and let $$ \chi^2 = \sum_{j=1}^r X_j^2,\qquad X_j:=\frac{(\nu_j - np_j)^2}{...
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Definition of heavy tails and moment generating functions

In this video (exact time already selected in the link) the connection between so-called 'heavy tails' and an infinite moment generating function is explained as follows: The benchmark to break into '...
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Lower bounds on the MGF for a mean zero random variable with variance $\sigma^2$

Let $X$ be mean-zero with variance $\sigma^2$. Is there a lower bound on the MGF for $X$ (or even simpler, $E e^X$) in terms of $\sigma^2$: $E[e^X] \ge f(\sigma^2)$? What about the general case where ...
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Show that there is no random variable $X$ such that the m.g.f. of $X$ satisfies $M_X(1) = 3$ and $M_X(2) = 4$

Show that there is no random variable $X$ such that the moment generating function of $X$ satisfies $M_X(1)=3$ and $M_X(2) = 4$. I'm not sure how to prove this, but I'm trying to go through known ...
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Find the MGF of a mixture distribution

I need to define the moment generating function of a mixture distribution. In particular I have a mixture of dirichlet-multinomial defined as $\sum_{i=1}^K \pi_i DM(\boldsymbol{\alpha},m)$ where $\...
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Moment generating function probability calculation

Let's say m.g.f (moment generating function) of a random variable $K$ has given by $𝜑(𝑡)=3/(3−𝑡)$ . Find $𝑃(K≤5)$.
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What's the meaning of ratio of second moment over third moment [closed]

The random variable $X \sim D$, what's the meaning of this term? $$E[X^2]/E[X^3]$$ *$E[X^3]$ is the third raw moment.
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Using MGF Technique to prove a theorem

Use the MGF technique to prove the theorem: If X1, X2,...,Xn are independent random variables where Xi~Normal(µi, σi2), then Y=Σ aixi follows a normal distribution with parameters µ = Σaiµi and σ2= ...
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What is the distribution of the m.g.f. that corresponds to the limit. [duplicate]

Let $\lambda>0$ be a fixed number. Suppose $X_n\sim Bin(n,\frac{\lambda}{n})$, for $n\in\mathbb{N}\setminus\{0\}$. Find the moment-generating function $M_{X_n}(t)$ for $X_n$. Compute for $t$ fixed,...
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56 views

Gradient of joint moment generating function of multivariate normal distribution

I'm trying to find the gradient of the joint MGF of the multivariate normal distribution, $M_{\pmb{X}}(\pmb{t}) = \exp\left[\pmb{t}^\mathsf{T} \pmb{\mu} + \frac{1}{2}\pmb{t}^\mathsf{T}\Sigma \pmb{t}\...
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104 views

Moment generating function of continuous random variable

I got a problem that said "find the Moment-generating function of $$f(t)=\frac{1}{4}e^{-t/4}\mathbb{I}_{(0,\infty)}(t)$$ And I solved it like this but I don't know if this' right $$\begin{align*}&...
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1answer
29 views

Finding the moment generating function of a random variable $Z$.

Given the m.g.f $M_x=\exp(3t+8t^2)$ of a continuous random variable $X$, find an m.g.f of the random variable $z=0.5(x-3)$, and use to find the mean and variance of $z$ My working: We know that $\mu=(...
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1answer
66 views

Is it possible to use MGFs for find the distribution of $X/(X+Y)$ when $X$ and $Y$ are independent and gamma distributed?

Suppose that $$Z=\dfrac{X}{X+Y}$$ $$X \sim Gamma(a,\lambda)$$ $$ Y \sim Gamma(b,\lambda)$$ with $X$ and $Y$ independent. I would like to see if it might be possible to determine the distribution of $...
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45 views

Computing expected values of a random variable, where we only have the moment generating function

This is problem 37 from chapter 4 of Bertsekas and Tsitsiklis's Introduction to Probability. The problem appears to come down to a filling of $n$ bins with $k$ objects and asking for the expected ...
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42 views

Question: Evaluating the moment generating function at a particular value.

I was reading about the moment generating function and I understand how i can find any moment with it. I was wondering however what happens if i evaluate the function at a particular value in its ...
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46 views

$n$-th derivative of $e^{x^2}$ at $x=0$ to calculate central moments of Normal distribution

I'd like to calculate $n$-th central moment of a normal random variable $X \sim N(m, s^2)$. There are 2 ways to do so: either calculate the integral (by definition of the expected value), or take the $...

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