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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Why finite sub-gaussian norm indicates sub-gaussian distribution?

The sub-gaussian norm of a random variable $\xi$ is defined as follows: $$\|\xi\| = \inf \left\{\lambda > 0 | \mathbb E\left(e^{\xi^2 / \lambda^2}\right) \leq 2\right\}.$$ When the moment ...
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33 views

Moment generating function of $f(x)=1/c$

Moment generating function of $f(x)=1/c$, $0<x<c$. Well, I know the calculation is easy. $M_X(t) = \frac{1}{ct} (e^{tc}-1)$. This question is Exercise 2.30 (a) Statistical Inference Book by ...
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1 answer
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How do I find the second order moment with a MGF?

Let $X$ be a random variable such that $M_X(t)=e^t M_X(-t)$. Find $E(X)$ and $E\left(X^2\right)$. I know the general procedure that to find the $n$th moment, the $n$th order derivative needs to be ...
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1 answer
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Derivation of moment generating function of a mixture of random variables

Question Let $Z$, $X$ and $Y$ be random variables, where $Z$ is formed by choosing a sample from $X$ with probability $q$ or a sample from $Y$ with probability $1 - q$. If $X$ and $Y$ are independent ...
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Is it possible to derive the moment generating function?

Let $X$ be a random variable such that $M_X(t)=e^t M_X(-t)$. Find $E(X)$ and $E\left(X^2\right)$. I know the general procedure that to find the expected value, the first order derivative needs to be ...
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2 answers
78 views

If random variable $X$ has moments of all orders, does this imply that the moment generating function of $X$ exist?

Suppose $EX^n<\infty$ for any $n$, does this imply that $M_X(t)=Ee^{tX}<\infty$ for all $t$ in some neighborhood of zero, that is, the moment generating function of $X$ exist? If this is not ...
3 votes
1 answer
33 views

Closed form for integer moments of Poisson random variable?

I'm interested in a closed-form (special functions included) for the integer moments of the Poisson distribution. Let $X\sim\operatorname{Poisson}(\lambda)$. Then we have for the moment generating ...
1 vote
0 answers
27 views

Let $\{X_i\}$ are i.i.d.. Given $Y$, how to estimate the probability of $n$ such that $S_n \leq Y < S_{n+1}$, where $S_n = \sum_{i=1}^n X_i$?

Let $\{ X_{i} \}_{i \in \mathbb{N}^{+}}$ are i.i.d. random variables with $\mathbb{E}(X_i) = \mu > 0$ and $\mathbb{E}(X_i^2) = \sigma^{2} > 0$.. Let $S_{k} = X_{1} + X_{2} + \cdots + X_{k}$. ...
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Finite moment generating function near zero but not subexponential

A centered random variable $X$ with moment generating function (MGF) $M_X(t) := \text E[e^{tX}]$ is subexponential if $\log M_X(t) \leq ct^2$ for some $c > 0$ on some neighborhood $(-\delta, \delta)...
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Existence of the moment generating function for a discrete uniform distribution

Problem: A random variable $X$ is said to have a discrete uniform distribution over $[1, N]$, with probability mass function as \begin{split} P(X = x) = \begin{cases} \frac{1}{N} &\text{...
1 vote
1 answer
28 views

$X_i \sim \text{Gamma}(\alpha, \beta)$ for $i=1,\dots, n$ where $\alpha, \beta >0$. Finding the pdf for the random variable $\frac{1}{n}\sum_i X_i$.

$\newcommand{\G}[1]{\text{Gamma}(#1)} \newcommand{\a}{\alpha} \newcommand{\b}{\beta} \newcommand{\rd}[1]{\mathrm{d}#1}$ ${\bullet \textbf{ Basics for the question:}}$ Let $X_i$ be i.i.ds (independent ...
2 votes
1 answer
38 views

Derivative of a moment generating function

Okay so I am assuming that you know what is the expectation of a random variable so let me define the Moment generating function of a random variable Let $X$ be a random variable whose probability ...
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How to result in moment generating function of bivariate weibull distribution FGMBWD?

from here https://link.springer.com/article/10.1007/s40745-019-00197-5 The pdf of a FGMBW distribution is defined as pdf To prove the moment generating function start with MGF the result as follows ...
1 vote
1 answer
36 views

Finding the first and third moment using MGF

Consider the PDF $$f(x) = 3x^2, 0≤ x ≤ 1$$ I'm trying to get the first and third moment from this PDF. I found the $E(X)$ using the usual formula to be $3/4$. However, I found the MGF to be $$\varphi(...
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1 answer
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Confused with the properties of the Moment generating function

Exercise 5 from "Probability and Statistics, 2nd edition, by Morris H. DeGroot" book. Suppose that $X$ has a uniform distribution on the interval $(a,b)$. Determine the moment generating ...
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2 votes
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Asymptotics of moment generating function

Consider r.v. $\xi$ with known c.d.f. $F$ and p.d.f. $f$. Let the corresponding moment generating function $M(z)$ be finite for all $z \in \mathbb{R}$. I am interested in deriving the asymptotics of $...
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Deriving Distributions and Covariance from Joint Moment Generating Function

Q: Suppose random variables 𝑋 and 𝑌 have joint moment generating function: What are the distributions for X and Y and their covariance, Cov[X,Y]? I am unsure how to derive the individual ...
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Moment generating function of random sample of Gamma Distribution

Let $Y_1,...,Y_n$ a random sample from the Gamma Distribution, $f(y_i|\gamma)=\dfrac{\gamma^3}{\Gamma(3)}y_i^2e^{(-y_i \cdot\gamma)}$, $y_i>0,\gamma>0$. Compute the momento generating function $...
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2 votes
2 answers
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Find two dependent random variables $X$ and $Y$ such that $\phi_{X+Y}(t)=\phi_{X}(t) \phi_{Y}(t)$, $\forall t$.

Question: Find two dependent random variables $X$ and $Y$ such that $\phi_{X+Y}(t)=\phi_{X}(t) \phi_{Y}(t)$, $\forall t$. I am having troubles with this question and am looking for some help. Here ...
2 votes
3 answers
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Given the CDF $F(x)=1-e^{-x^2};0≤x<∞$. Derive the moment generating function of $X$

I know you use $∫e^{tx}f(x)\,dx$ but I cant figure out the calculation. Any help would be appreciated
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Mean of exponentially tilted distribution

Let $X$ be a random vector in $\mathbb{R}^n$ with density w.r.t. the Lebesgue measure $f$. Assume that, for all $\lambda \in \mathbb{R}^n$, the moment generating function of $\mu$ is finite: $$M(\...
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moment generating function of a random variable dependent on a different one

Assuming I have a random variable $Y \sim \mathrm{U}(0,1)$, and a random variable $X$ has conditional distribution dependent on $Y$: $X\mid _{Y=p} \; \sim \mathrm{Bin}(n,p)$ (Bernoulli with success ...
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1 vote
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MGF of n random variables

I'm trying to evaluate the distribution of product of n iid normal random variables with non-zero mean firstly calculating the MGF of the product. I don't know if this method is suitable but C.Craig ...
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2 votes
1 answer
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High-order moments for a Gaussian random variable

I am currently reviewing the book "State Estimation For Robotics" by Timothy D. Barfoot ( document uploaded by the author http://asrl.utias.utoronto.ca/~tdb/bib/barfoot_ser17.pdf). The ...
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1 answer
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About moment generating function

It's on P91, probability theory and examples, version 5, by Rick Durrett. Although it's a section about large deviation, I think my question is universal for moment generating functions. Define $\...
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What happens to skewness as you play the same gamble multiple times?

Suppose there is a probability distribution $p$, over money outcomes. Suppose one draws $N$ times from $p$, with replacement, and takes the average money outcome over these $N$ draws. Call this ...
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Is a real random variable's moment-generating function the Laplace-Stieltjes transform of its CDF?

Wikipedia gives that the moment-generating function for a real random variable $X$ with cumulative distribution function $F$ equals $M_x(t) = \int \limits_{-\infty}^\infty e^{tx}\, dF(x)$, using the ...
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1 vote
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Under what conditions is $K^{(2)}(t) $ continuous at $t=0$ where $K(t)$ is the cumulant generating function?

For a random variable $X$ let \begin{align} K(t)= \log (E[e^{tX}]) \end{align} This quantity is known as the cumulant generating function. Question: What is some weak sufficient condition for \begin{...
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nth moment about mean of the binomial distribution

I was reading it from a book which solved it using the moment generating function:- $M(t)=e^{-tnp}(pe^t+q)^n=(pe^{qt}+qe^{-pt})^n$ after expanding it:- $M(t)= (1+\frac{pqt^2}{2!}+\frac{pq(q^2-p^2)t^3}{...
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Bochner-Type Theorem for MGF

Bochner's Theorem characterizes the the Fourier transform of a positive finite Borel measure on $\mathbb{R}$. This tells us, in principle, which functions can be characteristic function of a random ...
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35 views

Characterizing normal distribution by moment generating function

Let $X$ be a random variable. We know that if for all $\lambda \in \mathbb{R},E[e^{\lambda X}]=e^{\frac{\lambda^2}{2}},$ (from the properties of moment generating functions) then $X$ follows the ...
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1 vote
1 answer
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Determining a probability function for a sum of N i.i.d. geometric distributions between where N is discrete with a geometric pf [closed]

I was completing revision for an upcoming task and this question was presented. Was hoping for some insight! The random variable $N$ is discrete, with probability function $$f_N(n)=\begin{cases} p(1-p)...
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1 vote
1 answer
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Deriving recursion formula for the moments about the mean of the Poisson distribution

I'm having trouble seeing how this recursion formula holds. Let $a_x(\lambda) = (x - \lambda)^r \frac{\lambda^x e^{-\lambda}}{x!}$ Firstly, we have $$\log a_x(\lambda) = r \log(x - \lambda) + x \log \...
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Moment generating function for entry time of Brownian Motion

Let $B_t$ be a (standard) Brownian Motion and $\tau$ the stopping time $\tau = \inf \{ |B_t| \geq a \}$ for some $a >0$. Now I am interested in \begin{align} \mathbb{E}(e^{-\lambda \tau}) \end{...
1 vote
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Can a probability generating function fail to be analytic on any neighborhood of 1?

Any probability generating function $G(z)$ is analytic on at least the open unit disk of the complex plane, because its Taylor series expansion about $z=0$ has a radius of convergence of at least 1. ...
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1 vote
1 answer
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Which law has the random variable X if it has generating function $G_X(t)=a(3+2t^2)^3$?

Which law has the random variable X if it has generating function $G_X(t)=a(3+2t^2)^3$? First of all, what I did was try to find the value of $a$. We must have that $G_X(1-)=1\iff \lim_{t\rightarrow1^-...
3 votes
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Recurrence relations for even orthogonal polynomials

I have been playing around with the theory of orthogonal polynomials, and it occurred to me that we might be able to build a family of orthogonal polynomials from even powers of a variable $x$. For ...
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1 vote
1 answer
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Show that chi-squared random variables are sub-exponential with parameter $4$

Let $X \sim N(0, 1)$ so that $Y = X^2$ is distributed as chi squared with mean $1$. So the moment generating function of $Y$ is: $$\frac{e^{-\lambda}}{\sqrt{1-2\lambda}}$$ $Y$ is subexponential if the ...
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2 votes
3 answers
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Unsure about result involving Fibonacci Numbers $\sum_{n = 0}^\infty F_n$

I just starting working through a text on Moment Generating Functions. One elementary example is that of the Fibonacci Sequence and determining it's Moment Generating Function. I've found a result for ...
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1 vote
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Moment Generating Function of Y=2^(-X), where X ~ Geom(1/2)

My End Goal To compute the following Expected value: $$ \DeclareMathOperator{\E}{\mathbb{E}} \E \;\left(\frac1{\sum\limits_{i=1}^{B} 2^{-X_i}}\right)$$ where: $B$ is a constant (e.g., $B=4096$) $X_i \...
1 vote
1 answer
50 views

Consistent estimator of function of moments up to given order

Asume we the moments of a distribution exist up to order $n$. Consider the estimator $$t (x)=f(m_1(x),..., m_n(x)) $$ where $m_k(x):=1/r \sum_{i=1}^r x^k$. We assume we have $r$ samples of the ...
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Expected value of powers of Brownian Motion

At the moment I am following a uni course on Financial mathematics, the current subject is Brownian Motion. A subject I have now encountered a couple of times which I don't really understand is the ...
2 votes
1 answer
53 views

If $X$ is non-negative and bounded, then $ \mathbb E[e^{-X}] \le \frac{1}{e\mathbb E X}$

I guess this result is true, so I'm trying to prove it. Could you have a check on my attempt? Let $X$ be a non-negative bounded random variable. Then $$ \mathbb E[e^{-X}] \le \frac{1}{e\mathbb E X}. $...
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1 vote
1 answer
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Moment Generating Function of a summation of random variables where the upper limit is also random

How do we compute the Moment Generating Function of Q(t) here? I understand that we can use Wald's Equation to compute E[Q(t)]. Is there any theorem which can help me solve for the Moment Generating ...
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Using CDF to upper bound MGF

I have some random variable with a complicated but known analytical CDF and I need an upper bound on the MGF. Direct MGF computation is intractable due to the messy nature of the CDF so I was trying ...
0 votes
1 answer
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Joint MGF for Poisson

Suppose $X_i$ ~ Poisson$(λ_i ), i=1,2,3$. The $X_i$'s are independent. Let $Y_1 = X_1 + X_2$ and $Y_2 = X_2 + X_3$. Find the joint MGF and the covariance using the properties of expectation. Here I ...
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1 vote
1 answer
39 views

Find Moment Generating Function from a non linear function

Let $X$ be a r.v which has the following $mgf$ $$M_{x}(t) = (1-2t)^{-2}$$ If $Y = (100)(1.1)^{X}$ then calculate $E[Y]$ I know that $X \sim \chi^{2}_{(4)}$ but I am supposed to answer it only with $M_{...
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Thighter bound than Chebyshev for the error of an estimator

I have seen Chebyshev inequality applied to bound the error of an estimator wrt to its expected value. For example, the estimator of the mean of a set of values after they have been protected with a ...
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If $X$ and $Y$ are both independent standard normal random variables, then what is the distribution of $\frac{1}{2}\left(X^{2}-Y^{2}\right)$

As $X$ and $Y$ are standard normal variate, therefore: $$X^2 \sim \chi^2(1)\hspace{.5cm} \text{let's call it}\hspace{.5cm}\chi_X^2(1)$$ $$\text&$$ $$Y^2 \sim \chi^2(1)\hspace{.5cm} \text{let's ...
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1 answer
39 views

Uniquely determining marginal distributions

Let $X$ be a $n$-dimensional random variable with moment-generating function $$M(t) = \mathbb{E}[e^{\langle t, X \rangle}]$$ which we assume to exist for all $t \in \mathbb{R}^n$. Then we know that ...

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