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Questions tagged [moment-generating-functions]

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15 views

Why use the exponential in moment-generating function?

When trying to understand the moment-generating function, I've stumbled upon this general function: Mx(t) = E[e^tx], t ∈ R I understand that you have X = (X1, ... Xn) a series of random variables ...
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Finite Moments of Vector in Exponential Family

I am studying some notes on exponential families and there is a section on the computation of moments. The exponential family has the form $$\exp(\sum_{j = 1}^k \phi_j B_j(x) + C(x) - D(\phi))$$ I ...
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24 views

Sum of the Poisson distribution (my solution vs. textbook)

I feel something is wrong, but can't place it: Assume $X_i$ are i.i.d. Poisson distribution with parameter $\lambda$ and define $$Y = \sum_{i=1}^n X_i $$ $$M_X(t) = \exp((e^t-1)\cdot\...
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1answer
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Correlation and Moment generating function

Let $M=E[e^{t\cdot X}]$ be the moment-generating function of the random vector $X$ in $\mathbb{R}^n$. Then is it true that $$ E[(X -EX)^s] = \left. \frac{\partial \ln{M}}{\partial t^s} \right|_{t=0} $$...
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Branching process probability generating function

I'm trying to solve the following exercise but I can't seem to solve it. A branching process $(X_n :n \geq 0)$ has $P(X_0 = 1) = 1$. Let the total number of individuals in the first $n$ generations ...
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2answers
76 views

Compute the moment generating function of $Y = X_1X_2 + X_1X_3 + X_2X_3$

Suppose $X_1, X_2,$ and $X_3$ are independent and $N(0, 1)$-distributed. Compute the moment generating function of $Y = X_1X_2 + X_1X_3 + X_2X_3$. I know that any $X_iX_j$ with $i \not =j $ is a ...
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34 views

Variance of a sub-Gaussian random variable

For a zero mean sub-Gaussian R.V. we know that: $$ \mathbb{E}[e^{\lambda X}]\le e^{\frac{\lambda^2\sigma^2}{2}},\qquad\forall\lambda\in \mathbb{R}$$ From Taylor series expansion and equating the terms ...
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1answer
34 views

Integration by Substitution in $\int_0^{\infty}x^r\frac 1{\sqrt{2\pi}x}e^{-(\log x)^2/2}[\sin(2\pi\log x)]dx$

In Casella and Berger (2002) I found an example for non-unique moments (example 2.3.10 on page 64). They are providing the following 2 pdfs: $f_1(x) = \frac 1{\sqrt{2\pi}x}e^{-(\log x)^2/2}$, where $...
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Proof that the moment generating function of a lognormal distribution does not exist

In Casella and Berger (2002) I found a proof for the moment-generating function (mfg) of a lognormal distribution not being existent (see exercise 2.36 on page 81 and the answer provided here on page ...
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1answer
25 views

Ideas of how to find the distribution of Gamma(N,$\lambda$), with N~Geom($\alpha$)

I'm not exactly sure how to go about this, could I use the definition of the MGF or something along those lines?
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1answer
22 views

Sub-Gaussian and “nearly” sub-Gaussian random variables

Define $\Psi_{X}(\lambda) = \log E e^{\lambda X}$, and suppose that $EX=0$. We say that the random variable $X$ is sub-Gaussian with variance factor $v$ if: $$\Psi_{X}(\lambda) \leq v\lambda^{2}/2$$ ...
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1answer
32 views

method of moments and maximum likelihood estimators

I'm looking to find the estimate of $\mu$ for $n$ data using the method of moments and the maximum likelihood for the pdf given by $f(x) = \begin{cases} e^{-(x-\mu)}, & \text{if} \, x \geq \mu \\ ...
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1answer
22 views

Is the moment generating function smooth by definition?

So I'm going through Casella's Statistical Inference, and in Definition 2.3.6 he defines the moment generating function of a random variable $X$ with cdf $F_X$, denoted by $M_X(t)$, as $$M_X(t) = Ee^{...
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Method of moments when the first moment is $0$

I have a quick question regarding the method of moments estimator. Generally, when you have $k$ parameters you want to estimate, it suffices to find $k$ equations using $k$ moments. If you are ...
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1answer
21 views

What is the distribution of $P_M(M_B(t))$

$M_X(t)= P_M(M_B(t))$ $P_M(s)= (1-q+qs)^2 $ $M_B(t)= \frac{\beta}{\beta-t}$ Where P(x) is the probability generating function and M(x) is a moment generating function I identified M as $M \sim ...
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Moment Generating Function exercises: Knowing that $M_{X}(0)=1$ and $M´_{X}(0)= EX$

Hi guys, Any help with letter b of this exercise from Casella´s Book? I could finish the letter a). But I cant move in letter b). Any help? Thanks
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1answer
61 views

Assume that $Z_i\sim\mathcal{N}(0,1)$ are independent and prove that $\sum\limits_{i=1}^{n}(Z_i-\overline{Z})^{2}\sim\chi^{2}_{(n)}$

Assume that $Z_{i}\sim\mathcal{N}(0,1)$ are independent and prove the following results (a) $\displaystyle\overline{Z} = \frac{1}{n}\sum_{i=1}^{n}Z_{i}\sim\mathcal{N}(0,1/n)$ (b) $\overline{Z}$ and $...
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43 views

Moment generating functions and normal distributions

(a) If $X_{i}\sim\mathcal{N}(\mu_{i},\sigma^{2}_{i})$ are independent and $a_{i}\in\mathbb{R}$, $1\leq i \leq n$, then \begin{align*} Y = \sum_{i=1}^{n}a_{i}X_{i}\sim\mathcal{N}\left(\sum_{i=1}^{n}a_{...
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60 views

Given independent random variables $X_{k}\sim\text{Poisson}(\lambda_{k})$, what is the distribution of $X_{1} + X_{2} + \ldots + X_{n}$?

Show that, if $X_{k}\sim\text{Poisson}(\lambda_{k})$ and they are independent for $1 \leq k \leq n$, then \begin{align*} Y = \displaystyle\sum_{k=1}^{n}X_{k}\sim\text{Poisson}\left(\sum_{k=1}^{n}\...
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1answer
66 views

What is the moment generating function of the Laplace distribution?

Consider a random variable $X$ whose probability density function is given by $$f_{X}(x) = \frac{\lambda}{2}\exp(-\lambda|x-\mu|)\quad\text{for}\quad x\in\textbf{R}, \lambda > 0,\,\,\text{and}\,\,\...
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37 views

Calculate Probability from Moment Generating Function

Let 𝑋 and 𝑌 be two discrete random variables with the joint moment generating function $$𝑀_{𝑋,𝑌}(t_{1},t_{2})=(\frac{1}{3} e^{t_{1}} + \frac{2}{3})^{2} (\frac{2}{3} e^{t_{2}} + \frac{1}{3})^{3}...
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1answer
53 views

Poisson Variable with an Exponential Parameter becoming a Geometric Distribution?

Suppose Λ ∼ exponential(γ) and X ∼ Poisson(Λ). Use moment generating functions to show that X + 1 ∼ geometric(p) and determine p in terms of γ. In order to solve this problem, I first did: $E[e^{s(X+...
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1answer
47 views

Moment Generating function of one variable and Probability

I need only a hint please : Is there a way to calculate the probability if we have the moment generating function as in the following question: IF the moment generating function is given by : $$M(t)...
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0answers
35 views

Moment-generating function of $Z:=X_1X_2+X_3X_4$

Let $X_1,X_2,X_3,X_4$ be four indipendent random variable with normal distribution of mean 0 and variance 1. The exercise asks me to calculate the moment-generating function of $X_1X_2$. I was able to ...
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1answer
30 views

Finding moment generating function from a given probability mass function

Let $Y_1$ and $Y_2$ be two independent discrete random variables such that $p_1(y_1) = \frac13$; $y_1 = -2, -1, 0$ and $p_2(y_2) = \frac12$, $y_2=1,6$. Let K = $Y_1 + Y_2$. Find the moment ...
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0answers
40 views

Moment Generating Function - Cauchy Random Variable

I want to show that, for every $t\neq 0$, the moment generating function of the standard Cauchy distribution is equal to $+\infty$, i.e. $$M_X(t) = \int_{-\infty}^{+\infty} \frac{e^{tx}}{1+x^2}dx = +\...
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2answers
33 views

What is the difference between $E\{e^{-sx}\}$ and $E\{e^{sx}\}$ for MFG

I am working in wireless communication. When I cheek the books about MFG I found the MGF of random variable $X$ is given by the following formula $$ M_X(s)=E\{e^{sx}\} $$ However when I read ...
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28 views

Numerical approximation to Beta moment generating function

I have a Beta random variable $X \sim \text{Beta}(\alpha, \beta)$, and I'm interested in $\mathbb{E}[e^{2X}]$. The Beta distribution moment generating function is $$f(t) = {\displaystyle 1+\sum_{k=1}...
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2answers
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Moment-generating function of $m$ independent variables [closed]

Let $X_1,...,X_n$ be independent variables, each of them has a Discrete uniform distribution between $0$ and $m$, $m= \left( 2,3,4,,... \right)$. Let $Y$ be a random variable which is defined by ...
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1answer
42 views

Find the distribution of $Z=X+Y$ where both $X$ and $Y$ are exponentially distributed.

I have a problem where, in order to solve it, I need to find the distribution of $Z$. Say that $X\sim\text{exp}(\lambda)$ and $Y\sim\text{exp}(\mu)$. I don't want to use the convolution formula but ...
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27 views

How to find moments/compute this integral?

I have a steady state distribution which is of the form $$K[A+Bz]^{C}e^{D(1-z)}$$ where $A,\ldots,D$ are constants. I want to find moments of $z$. I do not know how I might go about it so that I can ...
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29 views

Squared Brownian motion and its moments

I have the following $X_t$ which satisfies: $X_t=a \cdot t+b \cdot W_t$ where $a,b \in R$ and $W_t$ is a Wiener process such that $W_t$ is normally distributed with $N(0,t-s)$ for $t>s$. Suppose ...
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1answer
58 views

Cumulants vs. moments

In high order statistics, what is the intuition for the difference between cumulants and moments? What does any of them measure and what is the intuition to use one of them over the other? ...
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1answer
36 views

What are the interest of the moments of a random variable?

Let $X$ a random variable. We define the moment of order $r\in\mathbb N$ by $m_r=\mathbb E[X^r]$. I know that the moment of order $1$ is the expectation, of order 2, one can get the variance, of order ...
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1answer
40 views

When is moment generating function finite on an interval?

I'm working on the following exercise: Let $X$ be a random variable on $(\Omega, \mathcal A, \mathbf P)$ and let $$\Lambda(t) := \log\left(\mathbf E\left[e^{tX}\right]\right) \quad \textrm{for all }...
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1answer
43 views

Probability that a random variable X belongs to the set of rational nos. [duplicate]

The question is a Multiple choice question Let $X$ be a random variable with the M.G.F. $$M_{X}(t) = \frac{6}{\pi^{2}}\sum_{n\ge1}\frac{e^{\frac{t^2}{2n}}}{n^2}\,,\;t\in R$$ Then $P(X \in Q)$, where ...
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0answers
24 views

Transformation of RV's Moments

Given a transformation of the RV $X$, how do it's moments transform? More Detailed Formulation Suppose that $X:(\Omega,\mathcal{F},\mathbb{P})\rightarrow \mathbb{R}$, is a random variable whose MGF ...
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0answers
27 views

Establish bound for a probability using moment generating function

I have the following question Let $X_{1}$, $X_{2}$, ..., $X_{n}$ be independent and identically distributed random variables with moment generating function $M_{X}(t)$, for -h < t < h, ...
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1answer
38 views

Computing the joint moment generating function for two functions of two random variables

Let $X$ and $Y$ be i.i.d random variables in the plane with a pdf $$f(x) = \frac{1}{\sqrt{2\pi}} \cdot \text{exp}(-x^{2}/2) \hspace{1cm} -\infty < x< \infty.$$ Let $U = X + Y$ and $V =...
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1answer
39 views

Moment generating function of a binary variable

We have a set of Random Variables $Y_i$ which takes the value $\alpha$ with probability $(1-p)$ and takes the value $1-\alpha$ with a probability of $p$. We have been tasked with finding the Moment ...
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1answer
43 views

Finding the moment generating function of $\min(Y,1)$

Let $Y\sim\text{Exp}(1)$ be a random variable. I denote the random variable $X$ as $X=\min(Y,1)$. The task is to find the moment generating function of $X$. By simply calculating the probability I ...
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1answer
29 views

Moment generating function of i.i.d

I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf I have two questions I know about moment generating function of ...
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2answers
34 views

Computing the joint moment generating function of two random variables

Let $P(N = k) = (1 - p)^{k - 1}p$, where $k = 1, 2, 3, \ldots$ and $0 < p < 1$. Let $X_{1}, X_{2}, X_{3}, \ldots$ be a sequence of i.i.d random variables with a common pdf $$f(x) = \...
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1answer
51 views

What conditions on the moments make a measure a probability measure?

For a positive Borel measure $\mu$ on the real line, let $\displaystyle{m_n = \int_{-\infty}^\infty x^n d\mu(x)}$, i.e. the $n$th moments of the measure. Are there any conditions on $m_n$ for when $\...
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1answer
20 views

Showing the sum of binomial independent variables follows a binomial distribution using moment generating functions

So I'm trying to solve the following problem: Show that if $X_i$ follows a binomial distribution with $n_i$ trials, and probability of $p_i=p$ for $i = 1,2,3...n$, and the $X_i$ are independent, then ...
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0answers
27 views

Laplace transform of generalized hypergeometric distribution

What is please the Laplace transform (moment generating function $M(t)$) of a generalised hypergeometric distribution shown below $$p_X(x)=K\cdot\frac{(a_1)_x\dots(a_p)_x}{(b_1)_x\dots(b_q)_x}\cdot\...
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0answers
13 views

What's the connection between moment generating function and methods of moment?

The moment generating function is given by $m(t)=E(e^{tY})$ where $Y$ is a random variable. However, the methods of moment is given by $m'_k=\frac{1}{n}\sum_{i=1}^nY_i^k$ where $\frac{d^k m(t)}{dt^k}...
2
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1answer
73 views

Given joint moment generating function (mgf), calculate $P(X + 2Y < 2X − Y)$

Given $X+2Y$ and $2X-Y$ are independent, and that $M_{X,Y}(t,u)=\exp\left[2t+3u+t^2+\dfrac{4}{3}tu+2u^2\right]$, how would one calculate $P(X + 2Y < 2X − Y)$?
2
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1answer
86 views

Given joint moment generating function, what value of $a$ makes $X + 2Y$ and $2X − Y$ independent?

I am new to joint moment generating functions and their properties, so am a bit stuck on how to begin the following problem: Given $M_{X,Y}(t,u) = exp[2t+3u+t^2+atu+2u^2]$, what value of $a$ makes $X ...
0
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1answer
28 views

Expectation property of moment generating function

Let $X$ be a non-negative random variable and define $M_X(t) := \mathbb{E}[e^{tX}]$. Suppose that $M_X(t) < \infty \mbox{ for all } t < \alpha$, where $\alpha > 0$. It can be shown that this ...