# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...