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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20 views

Derivation of Joint Moment Generating Function for Multivariate Normal [duplicate]

I'm attempting to derive the joint moment generating function for a multivariate normal, which takes the form below: $$M_{\mathbf{X}}(\mathbf{t})=E\left[e^{\Sigma_{j-1}^{n} t_{j} X_{j}}\right]=E\left[...
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41 views

Show that $E\exp(-tX_i) \leq \frac{1}{t}$

This is exercise 2.2.10 present in the book High-Dimensional Probability, by Vershynin. Let $X_1,\ldots,X_n$ be non-negative independent r.v with the densities bounded by $1.$ Show that the MGF of $...
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Method of Moment Estimation - Normal Distribution [closed]

NORMAL (GAUSSIAN) DISTRUBITION $X_1, X_2, . . . , X_n \stackrel{iid}{\sim} \mathcal{N}(\theta_1, \theta_2)$; Given : $θ_{1}$ is mean and $θ_{2}$ is variance $µ_{1}$=$M_{1}$= E($X_j$) = $θ_{1}$ $µ_{2}$=...
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Compound probability function and moment generating function

(Feller Vol.1, P.301) 4. Let $N$ have a Poisson distribution with mean $\lambda$, and let $N$ balls be placed randomly into $n$ cells. Show without calculation that the probability of finding exactly $...
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29 views

Properties of the moment generating function

Suppose $X \in L^1$ has a moment generating function $M(\theta) \equiv E(e^{\theta X})$ that is finite for every $\theta \in \mathbb{R}$ (so that $M'(\theta)$ exists for every $\theta$). It is ...
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31 views

Continuous Distribution and Probability density function Question [closed]

Suppose that $X_1, X_2, X_3$ denote a random sample of size three from a continuous distribution with probability density function (pdf) $f_X(x) = \frac{1}{\theta}e^{\frac{−x}{\theta}}, x>\theta.$ ...
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15 views

Properties of Legendre/Cramer's transformation of the moment generating function

Let $X \in L^1$ be a random variable on some probability space, define $M(\theta) \equiv E(e^{\theta X})$ as its moment generating function and let $D(M) \equiv \{\theta \in \mathbb{R} : M(\theta) <...
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157 views

Equality of Moment Generating Functions

Let $X,Y$ be be random variables whose moment generating functions $s\mapsto \mathbb{E}(e^{sX})$ exist and agree on either the interval $(-\delta,0]$ or on the interval $[0,\delta)$ for some $\delta &...
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39 views

Prove that $\frac{Y-E(Y)}{\sqrt{\operatorname{Var}(Y)}}$ converges in distribution to $Z\sim N(0,1)$ as $n\to \infty$

Let $X_i \sim \operatorname{Ber}(0.5)$ and $X_i$'s independent. Let $Y$ be a random variable with the same distribution as $\sum_{i=1}^n iX_i$. Prove that $\frac{Y-E(Y)}{\sqrt{\operatorname{Var}(Y)}}$ ...
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Which one is the correct mean formula of negative binomial distribution? $\frac{r}{p}$ or $\frac{(1-p) r}{p}$?

If we are looking to find the probability of observing the $6th$ head after $12$ independent flips and we let $X$ be the random variable for the number of flips of an unbiased coin I found that there ...
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Generating function for moments about mean

Find such a generating function for moments about the mean of the binomial distribution, and verify that the second derivative at $t=0$ is $n\theta (1-\theta)$. Till now, I have found the $$M_{X-\mu}(...
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126 views

Does proximity of moment generating functions implies proximity of characteristic functions?

Let's assume that $U$ and $V$ are non-negative random variables. Suppose that \begin{align} \sup_{t \ge 0 } \frac{| M_U(-t) - M_V(-t)|}{t} \le \epsilon \end{align} where $M_U(t)$ and $M_V(t)$ ...
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Determining moment generating function $\sum_{i=1}^n iX_i$

Let $X_i \sim Ber(0.5)$ and $X_i$'s independent. Let $Y$ be a random variable with the same distribution as $\sum_{i=1}^n iX_i$. Determine the moment generating function of $Y$. I figured the moment ...
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Moment generating function doubts

It´s a continous random variable. I have to get the MGF from a piecewise density function, but then, when I have to get the $\mathbb (x)$, the result is undefined, so I don't know if it's correct or I ...
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33 views

Showing that the $r^{th}$ derivative of the moment generating function is the $r^{th}$ raw moment

Is this a sound way of defining the $r^{th}$ derivative for the moment generating function $M_{X}(t)$—in order to show that when evaluated at $0$ it gives us the $r^{th}$ raw moment $E[X^{r}]$ ? ...
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Expectation of a binomial random variable raised to a non-integer power? (“non-integer moment” of a probability distribution)

Does anyone know if there's a closed-form expression of the expectation of the maximum likelihood estimator of a binomial variable raised to a non-integer power? Concretely, setting $$ P(m\mid N,p) = ...
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1answer
45 views

kth moment of an arbitrary function

I am given that $X\sim N(0,1)$ and that $Y=e^X$, and I am also given that the kth moment of Y is $E[Y^k]=E[e^{kx}]=M_X(k)=e^{\frac{k^2}{2}}$. This solution suggests that these relationships are self-...
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Is $X$ with moment generating function such that $\psi_X(1)=2$ and $\psi_X(2)=4$ constant?

I came across the following claim that puzzles me: Suppose that $X$ is a random variable with moment generating function $\psi_X$ such that $\psi_X(1) = 2$ and $\psi_X(2) = 4$. Show that $\...
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How to find the Moment Generating function of a function of random variables from their joint Moment Generating function?

Given the Joint Moment Generating function(MGF) of the random vector $X =(Y,Z)$ $$M_{Y,Z}(t_1,t_2) =e^{(t_1^2 + t_2^2 + t_1 t_2)/2} $$ How can I find the MGF of $Y+Z$ $Y-Z$ Is there any ...
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38 views

Moment generating function from probability mass function

We are given the pmf: $$f_X(k) = \frac{1}{k(k+1)}, k \geq 1 $$ and we have to compute the moment generating function. So far I've got: $$M_X(t) = E[e^{tx}] = \sum_{k=1}^{\infty} e^{tk} \frac{1}{k(k+1)}...
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Multifractal scalling exponent function \tau(q) for a Cauchy r.v.

The following function is called Multifractal scalling exponent function $\tau(q)$ $$ \tau(q)=\lim_{n\rightarrow\infty}\log_{n}\sum_{k=1}^{n}\left|\mathscr{H}\left(\left[\tfrac{k-1}{n},\tfrac{k}{n}\...
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Moment generating function for sum of independent random variables same as joint mgf

I'm seeing in general that for moment generating functions, the mgf of $X+Y$ where $X,Y$ are independent random variables is $M_{X+Y}(t) = M_X(t)M_Y(t)$. I'm also seeing that the joint mgf is given by ...
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How to compute the following (joint) mgf's for random variables (vectors) with the following (joint) pmf's

I have to compute the following (joint) mgf's for random variables (vectors) with the following (joint) pmf's, but I have no clue how it works. I have no background knowledge in Probability Theory and ...
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Bound sub-Gaussian variance proxy by variance for $[-1,1]$-valued random variables

Consider a random variable $X$ taking values in $[-1,1]$ with mean $0$ and variance $\sigma^2$. A quantity $V^2$ is a sub-Gaussian variance proxy if $\mathbb{E} \exp(\lambda X) \leq \exp(\lambda^2 V^2/...
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“Inverse” moment generating function of standard normal distributed random variable

This is just a trivial question maybe but, is the Moment generating function for $X$ the same as for $-X$ for a normally distributed random variable, so $E(e^{tX})=E(e^{-tX})$? If not, what is the ...
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53 views

Distribution of the product of a standard normal and uniform distribution

Given that $X\sim N(0,1)$ and $Z\sim \operatorname{Unif}(\{\pm1\})$. Prove that $Y = XZ$ is a standard Gaussian distribution. My approach: I started trying to find the pdf of them. I found that: $$...
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1answer
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Possible and Impossible Moment-generating function

$$M_1(t)=0.5(e^t + e^{-t}) \quad t\in R$$ $$M_2(t)=0.5(e^t - e^{-t}) \quad t\in R$$ $$M_3(t)=2*e^t - e^{2t} \quad t\in R$$ It's easy to see that all of the moments of M1 are exists, because all of ...
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45 views

Idenitity with moment generating function and complex integral

Let $\xi$ be a random variable with finite exponential moments. Define two functions$$C(k)=\Bbb E[(e^\xi-e^k)^+],\,k\in\Bbb R,\quad M(z)=\Bbb E[e^{z\xi}],\,z\in\Bbb C.$$ (a) Show$$M(z)=\int_{\Bbb &...
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1answer
25 views

Moment Generating Function of an Exponential variable

I know that when if we have an exponential random variable with parameter $\lambda$, the moment generating function is $\frac{\lambda}{\lambda-t}$ when $t < \lambda$, but what can I say about the ...
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Series of hypergeometric 2F1 functions / Moment generating function of random variable

Consider a random variable $X\in\mathbb N$ with probability mass function $$\Pr(X=k) = \frac{\binom{N+k+r-1}{r-1}}{\binom{N+r-1}{r-1}}p^k (1-p)^{N+r} {}_2F_1(N,k+N+r;k+N+1;p),$$ whereby ${}_2F_1$ ...
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57 views

Conditions for obtaining the characteristic function from MGF

Notation used is taken from Gallager's text on stochastic processes. For a random variable $X$, let $g_X(r)=\mathbb{E}[\exp(rX)]$ be its moment generating function where $r$ is a real number. ...
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25 views

Finding joint moment generating function of $Y_1$ and $Y_2$

Let $X_1$ and $X_2$ be independent standard normal random variables. Let $Y_1 = X_1 + X_2$ and $Y_2 = X_1^2 + X_2^2$. (a) Show that the joint moment generating function of $Y_1$ and $Y_2$ is $$\...
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1answer
21 views

Confusion about expectation / moment generating function / distributation

(I am currently studying a high dimensional probability course with very little background knowledge in probability theory as a whole, so I hope it's not annoying that I seem to be oblivious about ...
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Bound for the Moment Generating Function

In the answer for the question Construction of a MGF for conditional random variables the MGF is given: $M(\theta)= \left(\sum_{i=1}^n p_ie^{\theta a_i}\right)^A$, where $a_i \in R, i=1, \ldots, n$, $...
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Differences Between Characteristic Function and Moment-Generating Function

Why can't the moment generating function be defined for all random variables, while the characteristic function can be defined for all random variables?
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MGF Bernouli random variable

Question: If the sequence of moments of a random variable is given as $m_k = \frac{1}{2}(C^k+ (-1)^kC^k), c\in \mathcal{R}$, find the corresponding distribution. The given answer is that the random ...
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A bound on the moment generating function

Let $X$ be a random variable such that $|X|\le{K}$ for some $K>0$. I am trying to prove the following bound on the moment generating function of X: $$\mathbb{E}[\exp(\lambda{}X)]\le{}\exp(g(\...
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2answers
67 views

Moments do not characterize the distribution function

Let $X \sim N(0,1)$ and $Y = e^X$. Consider another random variable $Z$ and its PDF is $$f_Z(t) = [1 + \sin(2\pi \log_e(t))] f_Y (t)\quad 1\{t \ge 0\}\quad (1)$$ a) Show that $(1)$ is a valid PDF, ...
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1answer
36 views

Exploring Expectation Identity

Let $X\sim N(\mu ,\sigma^2)$ and g : $\Bbb R$ $\to$ $\Bbb R$ be a differentiable function and $E[|g'(X)|] < \infty$. a) Show that $E[g'(X)] = \sigma^{-2}E[g(X)(X-\mu)]$ b) Make use of the above ...
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1answer
25 views

MGF of a probability function [closed]

The MGF of p(Y=y) = Σ(from 0 to y) c(n,x) * p^x * (1-p)^(n-x) + c(n-y,y-x) * p(y-x) * (1-p)^(n-x).
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1answer
26 views

Determine the distribution of a function of variable through expectation of power functions

Let $X$ be a random variable (r.v.) defined on $D\subset{\mathbb{R}}$. Consider $Y(X)$ as a function of $X$, which can be treated as another r.v. Now suppose $$\mathrm{E}(Y^k) = \int_D Y^k(X)dF(x),$$ ...
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1answer
20 views

Find the moment generating function of $r_n Y_n$

Let $(Y_n)_{n\geq 1}$ be a sequence of geometrical random variables with parameter $r_n$ such that $r_n\to 0$ as $n\to \infty$. Find the moment generating function of $r_n Y_n$. You can ...
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2answers
84 views

Determining a random variable through the Taylor expansion of its moment generating function

Let $X$ be a random variable defined on a compact set $K\subset \mathbb{R}$. The moment generating function (MGF) of $X$, denoted as $M_X(t), t\in \mathbb{R}$, is defined as $$M_X(t) = \mathrm{E} [e^{...
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27 views

What strategy to use to compute this limit?

In a statistics problem, the following limit (on the left side) falls out from using various theorems on moment generating functions. The last step of the problem is to show that the limit on the left ...
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36 views

Density function from moment generating function

I am trying to obtain a probability density function (PDF) from its moment generating function (MGF) and the problem is the nature of the MGF. My MGF is $M(t)=I_o^{\beta}(s)$, where $\beta$ is a ...
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18 views

module of characteristic function <1

I have problems with this proof: How to prove that the module of characteristic function is less than one. Why $\vert\mathbb{E}[𝑒^{𝑖𝑡𝑋}]\vert\le \mathbb{E}[\vert𝑒^{𝑖𝑡𝑋}\vert]$?
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23 views

Expectation and moments inequality

X is a standard normal random variable. I came across this inequality in a paper --> $ | E[X^2] | \leq \frac{1}{2} E |X|^3$ ? I am not sure I understand how this works, can someone explain?
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1answer
22 views

Continuous Analogues of Generating functions

A generating function is a useful encoding of a sequence $\{a_n\}_{n=1}^\infty$. That is, given a generating function $A(x)$ we can find any $a_n$ by taking derivatives. Is there a continuous analogue ...
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2answers
32 views

Expectation of $E[e^{-X^2}]$ of a Gaussian distribution

Given Gaussian distribution $X \sim N(0, t)$, what is $E\left[e^{-X^2}\right]$? I used the Taylor expansion and Moment Generating Function to get this far: $E(e^{-X^2}) = 1 - E(X^2) + \frac{E(X^4)}{...
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0answers
36 views

Covariance of stochastic integrals - Poisson stochastic process -

I'm trying to improve my understanding on inhomogeneous Poisson processes. In particular, I'm focusing on stochastic integrals of the form $$I(T)=\int_0^Tf(t)dW_t$$ with $W_t$ arrival times of a ...

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