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Questions tagged [moment-generating-functions]

Description added to tag

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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
44 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
21 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
69 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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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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24 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
55 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
64 views

bound on the moment generating function [closed]

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
31 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})},...
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1answer
85 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_{...
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1answer
39 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 = \...
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93 views

Error in proof of MGF of standard half normal distribution?

I'm not getting the standard answer of but I'm not sure what's wrong with my reasoning. Any insight appreciated. Let $Z \sim N(0,1)$. We want to find the MGF of $\left|Z\right|$. By definition: $$\...
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2answers
79 views

MGF of sum of $N(t)$ iid random variables where $N(t)$ is a Poisson process

Given $(N(t),t\geq0)$ is a Poisson process with constant rate $\lambda\in\mathbb{R}$ (perhaps positive if required). Let $X_i$ be iid random variables that are also independent of the Poisson process....
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Why $m(t)=\frac{t^2}{1-t^2},t<1$,cannot be a moment-generating function?

$m(t)=\frac{t^2}{1-t^2},t<1$ The suggested answer provided by our teacher states that m(0)=0, so there's no real value with this function as mgf; hence mgf doesn't exist. Intuitively I understand ...
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Transformation of Moments

Suppose that $X,Y$ are random-variables and that $f$ is an infinitely-differentiable function with infinitely differentiable inverse, such that $$ f(X)=Y. $$ Question How are the moments of $X$ ...
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1answer
27 views

How to show a continuous random variable with even pdf also has a even moment generating function?

The moment generating function is $M_X(t)=$$\int_{-\infty}^{\infty} e^{tx}f(x) dx$ if X is continuous.
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1answer
50 views

Sum of Moment Generating Functions determined by a Random Variable

I have a question that is stated as follows: Two random variable $P$ and $Q$ have MGF's: $$M_P(s) = \left(\frac{1}{3} + \frac{2}{3}e^s \right) ^{10}$$ $$M_Q(s) = \frac{\frac{1}{5}4e^s}{1-\...
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2answers
32 views

Two quick notation questions about $M_X(t)$ the Moment Generating Function

The Moment Generating function of $X$, $M_X(t)$ is defined as $$M_X(t) = E[e^{tX}] = 1 + tE[X] + \frac{t^2 E[X^2]}{2!} + ... $$ Question 1 Is the MGF a function of one or two variables (as in $t$...
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151 views

Approximate values of $\operatorname{E}[\sqrt X]$ and $\operatorname{Var}[\sqrt X]$ for $X$ Poisson distributed with parameter $\lambda\to\infty$ [closed]

Assume that $X$ has a Poisson distribution with rate parameter $\lambda$. If $Y = \sqrt X$, using moment-generating functions or otherwise, show that $$\operatorname{E}[Y] \approx \sqrt\lambda - \frac ...
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$ \frac{p(1-p)e^t}{(1-p+p e^t)^2} \le \frac 1 4$: Use probability to prove or interpret.

While we can prove the following with AM-GM inequality, $$ \frac{p(1-p)e^t}{(1-p+p e^t)^2} \le \frac 1 4, t \in \mathbb R, p \in [0,1]$$ I want to find a proof that involves probability or ...
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5answers
80 views

how to show that $ \frac{\theta e^t(1-\theta)}{(1-\theta+\theta e^t)^2} \leq\frac{1}{4}$?

let $0 \leq \theta \leq 1$ , then how to show that $\forall t\in R$ $$ \frac{\theta e^t(1-\theta)}{(1-\theta+\theta e^t)^2} \leq\frac{1}{4}?$$ This is a step of a proof of hoffeding's lemma.
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Application of approximation of moments on poisson distribution

Exercises 101 in Chapter 4 Of the book "Mathematical Statistics and Data Analysis" by Rica states: Find the approximate mean and variance of Y = √ X, where X is a random variable following a ...
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76 views

Approximation method of moments (delta) method - worked out example

I am self-studying the delta method, but i cannot understand it. Here i provide an worked out example from Rice in hos book "Mathematical Statistics and Data Analysis". How are the calculations ...
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1answer
506 views

MGF for Binomial Distribution

I’m learning towards an exam I have. In one of the questions I've being asked to compute the MGF for the binomial distribution. My answer is slightly different from the official answer published by ...
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0answers
128 views

MGF of squared of inverse gaussian (IG) random variable

If x is an inverse Gaussian random variable, $x\sim IG(\lambda,\mu)$, i.e it is distributed by $f_x(x)={\sqrt{\lambda}\over\sqrt{2\pi x^3}} \exp(-\frac{\lambda(x-\mu)^2}{2\mu^2x}), \quad x>0$ ...
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1answer
263 views

Finding the moment generating function with a probability mass function

We have the probability mass function for a random variable $X$ given in table form: $$ \begin{array}{c|cccc} x & 1 & 2 & 3 & 4 \\ f(x) & 0.1 & 0.25 & 0.3 & 0.35 \end{...
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1answer
37 views

How to prove that $\mathbb E(x^n)-\mathbb E(x)^n \ge 0; \quad n \in \mathbb Z_+.$

How to prove that $$\mathbb E(X^n)-\mathbb E(X)^n \ge 0; \quad n \in \mathbb Z_+.$$ X is a R.V. in the range of $[0, 1]$ Any hints appreciated. Thanks.
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2answers
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Compute $m_Z (t)$. Verify that $m'_Z (0)$ = $E(Z)$ and $m''_Z(0) = E(Z^2)$

Let $Z$ be a discrete random variable with $P(Z = z)$ = $1/2^z$ for $z = 1, 2, 3,...$ (b) Compute $m_Z (t)$. Verify that $m'_Z (0)$ = $E(Z)$ and $m''_Z(0) = E(Z^2)$ $E(Z) = \sum_{z=1}^{\infty} z P(Z ...
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3answers
74 views

$X\sim \text{Exp}(\lambda)$ use the moment generating function ($m_X(t)$) to find $E(X)$ and $E(X^2)$

Q1) Let $X\sim\text{Exp}(\lambda)$. Find $m_X(t)$. My attempt: $$m_X(t) = E[\text{e}^{tX}] = \int_{0}^{\infty}\, \text{e}^{tx} \lambda e^{-\lambda x}\,\text{d}x = \int_{0}^{\infty}\,e^{-\lambda x + ...
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0answers
65 views

Find the first moment of a probability distribution governed by a nonlinear first order ODE

May I ask if there is any standard way to find the first moment of a probability distribution governed by a nonlinear first-order ODE. For example, $$ \frac{\mathrm d p(x)}{\mathrm dx} = \alpha(x) p(x)...
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2answers
1k views

Finding a Joint Moment Generating Function

Please consider the following problem and my solution to it: Problem: Let $(X,Y)$ be a continues bivariate r.v. with joint pdf \begin{eqnarray*} f_{XY}(x,y) &=& \begin{cases} e^{-(x+y)} & ...
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79 views

Finding the nth derivative of Poisson process MGF

Anyone know how to find the Expectation of a Poisson process in Lp by using moment generating function. In other words, the nth derivative of this mgf.
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1answer
22 views

The moments of the sum of variables following standard normal distribution divided by the sum of the squares of them

Let i.i.d. $t_i\sim N(0,1), i = 1,2,\dots,n$, and \begin{equation} X = \frac{\sum_i t_i}{\sum_i t_i^2}. \end{equation} How to calculate the first two moments of $X$, i.e., $\mathrm{E}(X)$ and $\...
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1answer
166 views

Linear combination of power law distributions

I am exploring whether a linear combination of power law distributions is also a power law distribution. Specifically, if $X \sim (\alpha -1) x_{\min}^{(\alpha -1)} x^{-\alpha}$ and $Y \sim (\beta -1) ...
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0answers
74 views

Moment Generating function with Poisson and Exponential distributed variables

what happens when I have the following MGF with the $N$ and $M$ independent: $$ E(e^{-\rho R N M}),\mbox{ with }N \sim \mbox{Poisson} (\lambda), M \sim \mbox{Exponential} \left(1/t,1/t^2\right).$$ A ...
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1answer
37 views

Existence of the MGF of $X_2$

For $\theta>0$, let $X_1, X_2,\ldots,X_n$ be an iid sequence of Uniform$(0, \theta)$ random variables and we set $X_{(n)}=\max\left(X_1, X_2, \ldots, X_n \right)$. I have already shown that the ...
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1answer
60 views

How to simplify $\sum _{l=1} ^{k} \sum_{j_1 \ge 1} \cdots \sum_{j_l \ge 1, j_1 + \cdots + j_l=k} \binom{k}{j_1 \cdots j_l} \binom{-2}{l} 2^l$

I'm trying to simplify the following equation. $$\sum _{l=1} ^{k} \sum_{j_1 \ge 1} \cdots \sum_{j_l \ge 1, j_1 + \cdots + j_l=k} \binom{k}{j_1 \cdots j_l} \binom{-2}{l} 2^l$$ Or more specifically, I'...
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1answer
46 views

Simplify and compute the MGF of $[1-(1-(1-e^{-ax})^{N_1})(1-(1-e^{-ax})^{N_2})]^{M-1} $

Let $X_1$ and $X_2$ two random variable with CDF \begin{align} F_{X_1}(x)&=(1-e^{-ax})^{N_1}\\ F_{X_2}(x)&=(1-e^{-ax})^{N^2} \end{align} Let $Z$ random variable with CDF $$ F_Z(z)=[1-(1-F_{...
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1answer
16 views

PDF for Moment Generating Function $(1 +\beta t)^{-\alpha}$ when $\beta > 0$

I'm trying to find the PDF for the following MGF. $$(1 + t/2)^{-3/2}$$ I already know that $(1 - \beta t)^{-\alpha}$ when $\beta > 0$ is an MGF for the Gamma distribution $\Gamma(\alpha, \beta$). ...
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2answers
78 views

Showing if $\frac{X}{c}\sim\text{Gamma}(a,b)$ then $X\sim\text{Gamma}(a,cb)$

I am trying to show that if $$\frac{X}{c}\sim\text{Gamma}(a,b) \ \ \ \ \ \ \text{then} \ \ \ \ \ \ \ X\sim\text{Gamma}(a,cb)$$ My first approach was to use a PDF transformation. I let $$Z=\frac{X}{...
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1answer
257 views

Variance using moment

In a group of 15 health insurance policyholders diagnosed with cancer, each policyholder has probability 0.90 of receiving radiation and probability 0.40 of receiving chemotherapy. Radiation and ...
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1answer
50 views

Show for $\ c>0\ $ that $\ cY\sim \ \text{Gamma}(\alpha,c\beta)$

Show for any constant $\ c>0\ $ that $\ cY\sim \ \text{Gamma}(\alpha,c\beta)$ $$Y\sim\text{Gamma}(\alpha,\beta)$$ $$f_Y(y)=\frac{1}{\Gamma (\alpha)\beta^\alpha}e^{\frac{-y}{\beta}}y^{\alpha-1} ...
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0answers
66 views

Using MGF's to characterize a distribution

Let $X_1,X_2,X_3$ be independent such that for all $x > 0$, $$ P(|X_i| > x) < e^{-x}, \;\;\; i = 1,2,3 $$ Prove that if $X_1+X_3$ and $X_2+X_3$ have the same distribution, then so ...
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2answers
110 views

Moment generating function of the average of two variables

I am given two variables $Y_1$ and $Y_2$ obeying an exponential distribution with mean $\beta= 1$ We are asked what the distribution of their average is and the solution must be found using moment ...
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0answers
102 views

Finite Exponential Moment

Consider $X_1, X_2, X_3$ ... random variables i.i.d. such that $P(X_i=1)=p$ and $P(X_i=-1)=1-p$. Consider the random walk $(S_n)_{n\ge 0} $ with $S_0=0$ and for $n\ge 1 $, $S_n = \displaystyle\sum^{...
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0answers
52 views

Delta method application

I have been trying to understand the delta method, with no success. Our professor gave us an exercise (with solution), however I do not even know where to start solving the problem. Question: ...
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4answers
740 views

Find the Variance of Negative Binomial Distribution via the MGF

Find the Var$(X)$ given that $$m_X(u)=\Big(\frac{p}{1-(1-p)e^u)}\Big)^r \ \ \ \ \ \ \ u<\text{ln}((1-p)^{-1})$$ I have found $\mathbb{E}(X)$ to be $$\mathbb{E}(X)=m'_X(u)$$ $$\mathbb{E}(X)=r\Bigg(\...
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1answer
89 views

Show $\lim_{p\to 0} m_Y(u)=\Big(\frac{1}{1-2u}\Big)^r$ where $Y=2pX$

For any $0<p<1$ and $r$ a positive integer, the probability function $$f(x)={{r+x-1}\choose{x}}p^r(1-p)^x \ \ \ \ \ \ x=0,1,2...$$ defines a random variable $X$. I have computed the mgf of ...
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1answer
342 views

Expectation and Variance of Negative Binomial Distribution from the MGF

For any $0<p<1$ and $r$ a positive integer, the probability function $$f(x)={{r+x-1}\choose{x}}p^r(1-p)^x \ \ \ \ \ \ x=0,1,2...$$ defines a random variable $X$. I have computed the mgf of ...
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1answer
1k views

MGF of The Negative Binomial Distribution

For any $0<p<1$ and $r$ a positive integer, the probability function $$f(x)={{r+x-1}\choose{x}}p^r(1-p)^x \ \ \ \ \ \ x=0,1,2...$$ defines a random variable $X$. Compute the mgf of $X$ to ...