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Questions tagged [moment-generating-functions]

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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
28 views

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
40 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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20 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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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
49 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
36 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
37 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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22 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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23 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
31 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
27 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
34 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
27 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
29 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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0answers
39 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 $\...
2
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1answer
15 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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23 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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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}...
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1answer
52 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 $P(X + 2Y < 2X − Y)$?
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1answer
82 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 ...
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1answer
27 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 ...
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1answer
26 views

Linear Combination of Independent Random Variables

I am working on a problem and am a bit stuck The problem: For independent random variables X ~ N(-1,3) Y ~ N(0,2) Z ~ N(4,1) Consider U:= 2X - 4Y - Z + 5 Find the Expectation, variance, and MGF ...
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1answer
20 views

Finding probability distributions associated with moment generating functions [duplicate]

I think the answer to my question is pretty simple, but I'm not able to figure it out. The question is: Find the distribution which corresponds to the moment generating function $\frac{2e^t}{3-e^t}...
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1answer
21 views

Find the Distribution that corresponds to the given MGF

I am working on a problem and am a little bit confused. I need to find the distribution that corresponds to the MGF: $2e^t\over3-e^t$ Do we need to separate this into something like: 2e$^t$ and $1\...
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1answer
15 views

Finding the Law of X

I am woking on a problem and am unsure how to approach it. It is finding the law of X (the distribution that correspond to) for MGF $2e^t\over3-e^t$ I am thinking that this is perhaps an exponential ...
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4answers
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Finding MGF of Multiple RV

I am working on a problem and am a bit stuck. The problem: Let X1, X2, X3 be i.i.d random variables with distribution P(X1=0) = $1\over3$ P(X1=1) = $2\over3$ Calculate the MGF of Y = X1X2X3 Not ...
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2answers
37 views

Finding Standard Deviation from MGF

I a working on a problem and am a little bit confused at how to approach solving it. The problem: Given the MGF F(t) = $1\over(1-2500t)^4$ Calculate the SD. Do we need to do some substitution with ...
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0answers
24 views

Moment generation function - compound Poisson

Let $S_1$ be distributed with compound Poisson $\lambda_1=2$ and discrete indeminizations: $f_1 (x), x \geq 0$. And let $S_2$ be compound Poisson distributed with $\lambda_2=4$, and discrete ...
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1answer
26 views

Moment Generating Function of a Continuous Random Function

I am stuck on this statistics problem: If the pdf of a measurement error $ X $ is $f(x)=0.5e^{-|x|}$, $ -\infty<x<\infty $, show that $ M_X(t)=\frac{1}{1-t^2} $ for $ |t|<1. $ I've split ...
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0answers
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Tight Bounds on Moments of $X = a_1x_1 + \cdots + a_nx_n$

Suppose that $x_i,z_i$ are i.i.d and that the moments $\mathbb{E}x_i^m \leq K \mathbb{E} z_i^m$ uniformly for $m=1,2,\ldots$. For simplicity's sake take $x_i$ and $z_i$ to be symmetric so we can ...
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1answer
30 views

Using MGFs to determine independence of sum and difference of two variables $(X+Y$ and $X-Y)$

I have a question for class that asks: Let $X$ and $Y$ be i.i.d. Unif$(0,1)$. Are $X+Y$ and $X-Y$ independent? In an earlier part of the question I found that the covariance between (X+Y) and (X-Y) ...
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1answer
34 views

Moment-generating function of a random variable with mean 0 and variance 1

Consider a random variable $X$ with mean $0$ and variance $1$. Given $0 < d < 1$, do we have $$E(e^{d X}) \leq 1 + O(d^2)$$ (it's true when $|X| \leq 1$)
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28 views

Lim sup of moments

In the page 122 of Durrett's book, with definitions $\mu_k=\int x^kdF(x)$ and $v_k=\int |x|^kdF(x)$, If $\text{limsup}\frac{\mu_{2k}^{1/2k}}{2k}=r<\infty$, then by using Cauchy-Schwarz $v_{2k+1}^2\...
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1answer
28 views

Distribution of a stable and exponential r.v

I am given an absolutely continuous positive r.v $X$, such that $\mathbb{E}[e^{-sX}]= e^{{-s}^\alpha}, \ s>0.$ The goal is to prove that $(Y/X)^\alpha$ is an Exponential r.v with intensity $1$, ...
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0answers
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Standard Normal Moments and Combinatorics

At around 16-17 mins in this video, the professor calculates the even moments of the standard normal. If $Z \thicksim N(0,1)$ then $$\mathbb{E}[Z^{2n}] = \frac{(2n)!}{2^n \cdot n!}.$$ The right hand ...
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Prove that return process follows log normal distribution

Let $S_k$ be asset price at time $k$, $\mu$: mean return, $\sigma$: asset volatility. The price process is: $ \frac{S_k}{S_{k-1}} = exp\{\sigma Y_k + (\mu - \frac{1}{2}) \sigma^2)\} $ where $Y_k \...
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1answer
17 views

Moment generating function is finite given limsup property

For a random variable $X$ define the moment-generating function $M_X(t) = \mathbb{E}[e^{tX}]$. If it is known that $$\limsup_{x\to\infty}\frac{\log\mathbb{P}(X>x)}{x} =- c < 0,$$ how can it be ...
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0answers
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Let X ∼ Expo(λ). You can assume you know that λX ∼ Expo(1), and that the nth moment of an Expo(1) random variable is n!. Find the skewness of X.

This is a question for class. My prof has given me some guidance but I can't wrap my mind around it. My prof said I should start by considering the equation for skewness very generically, and use the ...
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1answer
38 views

Why is moment generating function represented using exponential rather than binomial series?

A Moment Generating Function (MGF) of a random variable $X$ is represented by an exponential $$M_X(t) = E[e^{tX}]$$ Why isn't an MGF represented by binomial expansion? For instance, $$M_X(t) = E\...
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1answer
18 views

Moment generating function $Y$ [closed]

If $X$ is a random variable, normally distributed with unknown parameters how could I find the mgf of random variable $Y$, where Y=$e^X$? I am able to find mgf of $X$ from the mgf of a standard ...
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1answer
52 views

Finding the Probability from the sum of 3 random variables

Let $X_1, X_2$ and $X_3$ be three independent normal random variables having mean $\mu= 0$ and variance $\sigma^2=16.$ Compute $P(X_1^2+X_2^2+X_3^2>8).$ Hint: First transform the random ...
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0answers
40 views

Negative Binomal Random Variable Question

I am stuck with the question below. If X is a negative binomial random variable, then $$ Y=r+x $$ is the total number of trails necessary to obtain r S's. Obtain the moment generating function of Y ...
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0answers
18 views

Moment generating function of f(X)

I know that we can obtain the expected value of a random variable: \begin{align} E[X] = \int x \space p(x) \space dx \end{align} and the expected value of a function of this variable: \begin{align}...
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1answer
39 views

Minimize Chernoff Bound Exponential Distribution

For the distribution $f_X(x)=\lambda e^{-\lambda x} \ \ x =[0,\infty]$ I am trying to understand how to utilize the Chernoff Bound. The Chernoff bound is: $P(X>x) \leq g_X(r)e^{-rx}$ where $g_X(r)$ ...
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1answer
42 views

bound on the moment generating function

Let $X$ and $Y$ be discrete random variables. Is there a known class of joint distributions $p(x,y)$ which satisfies the following property: $$\mathbb{E}\left[ e^{\lambda X} e^{\lambda Y} \right]&...
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1answer
23 views

Prove the moment generating function of the random variable is the function of its eigenvalues.

I want to prove that $$ \mathbb{E} \Big\{ \exp\left(-x\lVert{\mathbf{H}}\rVert_F^2\right)\Big\} = \frac{1}{\det(\mathbf{I}_{m,n}+x\mathbf{R})}=\prod_{i=1}^{m\times n}\frac{1}{1+x\lambda_i(\mathbf{R})},...
3
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1answer
75 views

Bounding the MGF of a non-homogeneous Rademacher chaos of order two

I am trying to bound a quantity of the form $$ \mathbb{E}\left[ F\!\left( \sum_{i,j} a_{ij}\varepsilon_i\varepsilon_j' + \sum_{i,j,k} b_{ijk}\varepsilon_i\varepsilon_j\varepsilon_k'+ \sum_{i,j,k} b_{...
0
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1answer
30 views

What is the example of $X$ being in $L_1$ but mgf of $X$ being infinity? [closed]

I have spent quite a few hours trying to come with the example that shows the following: "There exists a non-negative random variable $X$ such that $X$ is in $L_1$ (integrable), but mgf of $X = \...