Questions tagged [moment-generating-functions]

Description added to tag

15
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2answers
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Distribution of the difference of two normal random variables.

If $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference? I will present my answer here. I am hoping to know if I am right or wrong. ...
14
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2answers
29k views

Deriving Moment Generating Function of the Negative Binomial?

My textbook did the derivation for the binomial distribution, but omitted the derivations for the Negative Binomial Distribution. I know it is supposed to be similar to the Geometric, but it is not ...
11
votes
1answer
266 views

About the “Cantor volume” of the $n$-dimensional unit ball

A simple derivation for the Lebesgue measure of the euclidean unit ball in $\mathbb{R}^n$ follows from computing $$ \int_{\mathbb{R}^n}e^{-\|x\|^2}\,dx $$ in two different ways. See, for instance, ...
9
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1answer
3k views

Tail bounds for maximum of sub-Gaussian random variables

I have a question similar to this one, but am considering sub-Guassian random variables instead of Gaussian. Let $X_1,\ldots,X_n$ be centered $1$-sub-Gaussian random variables (i.e. $\mathbb{E} e^{\...
9
votes
1answer
1k views

What is the meaning of the cumulant generating function itself?

If we define the characteristic function for a random variable X as $\Phi(t)=<e^{itX}>$ then it seems like we can think of it as essentially a spectral decomposition that measures the ...
9
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1answer
6k views

How to find probability distribution function given the Moment Generating Function

After searching, I found two questions like mine, but didn't see my answer to my question. Finding a probability distribution given the moment generating function Finding probability using moment-...
8
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0answers
695 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ $$\...
7
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2answers
6k views

Finding the moment generating function of the product of two standard normal distributions

The following question is on my homework assignment that I cannot figure out: Let U and V be independent random variables, each having a normal distribution with mean zero and variance one. Find ...
7
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0answers
602 views

Uniform convergence of Empirical Moment Generating Function

In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as: $$ \begin{...
6
votes
3answers
45k views

Moment Generating Function of Poisson

I'm unable to understand the proof behind determining the Moment Generating Function of a Poisson which is given below: $N \sim \mathrm{Poiss}(\lambda)$ $$ E[e^{\theta N}] = \sum\limits_{k=0}^\infty ...
6
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2answers
5k views

If $X$ is normal, is $\exp(X)$ still normal? How to find its mean and variance?

$X$ is a random variable for normal distribution: $X\sim N(\mu, \sigma^2)$. What is the mean and variance of $\exp\{x\}$? My attempt: $$E[\exp\{x\}]=\exp \{E[x]\} \text{, by the invariance property?}...
5
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3answers
1k views

Moment Generating Function and Inverse Laplace transform

I need to compute the inverse Laplace transform of the function $$ M(t)=e^{\frac{t^2}{2}} $$ Now, I know that this is a normal distribution with mean zero and variance 1, but how the computations are ...
5
votes
2answers
359 views

Asymptotic Moments of the Binomial Distribution, $E(X/(np))^k = 1 + O(k^2/n)$?

Let $X \sim \text{Binomial}(n, p)$ be the sum of $n$ Bernoulli($p$) random variables. What is the value of $E(X/(np))^k$, where $k$ is a large integer, as $n$ grows large? From calculations the ...
5
votes
1answer
2k views

Derivative of moment generating function

If the moment generating function of $X$ exists, i.e., $$M_X(t)=E[e^{tX}],$$ then the derivative with respect to $t$ is usually taken as $$\frac{dM_X(t)}{dt}=E[Xe^{tX}].$$ Usually, if we want to ...
5
votes
1answer
302 views

Finding the mgf when moments are given

Let $X$ be a random variable and it is known that the mgf of $X$ exists. If the $k$th moment is given by $m_k=\mathbb E[X^k]=\frac{(2k+1)!}{k!2^k}$ for $k=0, 1, ...$ Problem: Find the mgf of $X$. ...
5
votes
1answer
123 views

Sorting out some integrals from physics

I'm doing some physics for a change, and I'm trying to sort things out a bit. From the definitions of mass, torque, momentum and angular momentum I've come up with the following integrals: \begin{...
5
votes
1answer
117 views

Nonzero solutions to $\mathbb E\left[e^{\theta X}\right] = 1$?

Suppose $X$ is a random variable with $\mu=\mathbb E[X]\ne0$ and that $X$ has a finite moment generating function on some open interval containing $0$. Then for what $\theta\ne0$ does the following ...
5
votes
0answers
408 views

Moment generation function -> characteristic function uniqueness

Here's my proof that moment generation function (if exists) uniquely determines characteristic function. Can you please see how to make it more rigorous or improve in either way (e.g. by citing ...
4
votes
4answers
901 views

How to integrate: $\int_{0}^{\infty}e^{tx}(x^2e^{-x})/2dx$

I'm working on a few moment generating function problems and I came across: $f(x)=(x^2e^{-x})/2$ for $x>0$, and zero otherwise. Find the mean. The mean is the same as the expected value. If we ...
4
votes
2answers
5k views

Sum of independent Poisson random variables is a Poisson random variable

Suppose $x_1$ and $x_2$ are independent Poisson random variables with parameters equal to $\lambda_1$ and $\lambda_2$ respectively. Show the sum of $x_1$ and $x_2$ is also a Poisson random variable ...
4
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2answers
3k views

Expected value of the Max of three exponential random variables

So the question asks: Let $X_1,X_2,X_3\sim \operatorname{Exp}(\lambda)$ be independent (exponential) random variables (with $\lambda> 0$). (a) Find the probability density function of the ...
4
votes
1answer
475 views

Moment generating function and probability

Problem:Let $X$ and $Y$ be identically distributed independent random variables such that the moment generating function of $X +Y$ is $$M(t) = 0.09 e^{−2t} + 0.24 e^{−t} + 0.34 + 0.24 e^t + 0.09 e^{2t}...
4
votes
2answers
124 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 ...
4
votes
1answer
930 views

What is Moment Generating Function simply explained?

I'm taking Econometrics this Trimester and there a quite few things I don't get yet, and one of them is MGF. I have tried Wikipedia etc. but seems to be written in a bit more advanced language. Any ...
4
votes
2answers
197 views

$\mathrm{E}[e^{u X}|\mathcal{A}] = \mathrm{E}[e^{u X}|\mathcal{B}]$ implies equality of conditional distributions

Let $X$ be a random variable on the probability space $(\Omega, \mathcal{F}, P)$ and $\mathcal{A} \subset \mathcal{B} \subset \mathcal{F}$ be a $\sigma$-subalgebras. I want to prove that if $$ \...
4
votes
2answers
55 views

Prove that $\lim_{y\to\infty}\frac{y}{t-1}e^{y(t-1)} = 0$.

For the distribution $g(y)=ye^{-y}$ for $y\geq 0$ and $0$ otherwise, I have shown, through integrating by parts, that the moment generating function is: $M_Y(t)=\int_0^\infty e^{ty}ye^{-y}dy=\left[\...
4
votes
1answer
1k views

Find the moment generating function of the sum of exponential random variables $S=X_1+X_2+X_3+X_4$

Let $X_1+X_2+X_3+X_4$ be iid exponential random variables with parameter λ, and $S=X_1+X_2+X_3+X_4$ S follows the gamma distribution with parameters $\lambda$ and $r=4$. We know that an exponential ...
4
votes
1answer
303 views

History of Moment Generating Functions

I am beginning to appreciate how important Moment Generating Functions (MGFs) are regarding various common probability distributions and the ways their expectations/variances are calculated. My open-...
4
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1answer
1k views

Find m.g.f. given $E(X^r)$ function?

"Let $X$ be a random variable with $E(X^r) = 1 / (1 + r)$, where $r = 1, 2, 3,\ldots,n$. Find the series representation for the m.g.f. of $X$, $M(t)$. Sum this series. Identify (name) the probability ...
3
votes
2answers
10k views

Determine the PDF from the MGF [closed]

If the moment generating function is given as; $ \psi_X(s) = e^{s^2}$ How can i determine the PDF of $X$?
3
votes
2answers
3k views

convexity of log of moment generating function

Why is log of a moment generating function of random variable Z is convex? that is $\log \mathbb{E}[\exp(\lambda.Z)]$ My logic says since expectation is linear so it is in particular convex and ...
3
votes
2answers
166 views

Showing inequality: $pe^{x(1-p)}+(1-p)e^{-xp} \leq e^{x^2(3/4)p}$ for $0 \leq p \leq 1/2, 0 \leq x \leq 1$?

How can I show that $$pe^{x(1-p)}+(1-p)e^{-xp} \leq e^{x^2(3/4)p}$$ for $0 \leq p \leq 1/2, 0 \leq x \leq 1$? I've been stuck on this for a long time; I tried expanding out the taylor series on ...
3
votes
1answer
605 views

Characteristic function and moment generating function: differentiating under the integral

In order to justify the interchange of the derivative and integral when differentiating a characteristic function, one can use the dominated convergence theorem: $$\frac{d}{dt} \int e^{itx} P(dx) = ...
3
votes
3answers
7k views

$X$ standard normal distribution, $E[X^k]=?$

I'm stuck with a homework problem where we are supposed to prove that the expected value $E[X^k]$, if $X$ has standard normal distribution, is equal to: $$E[X^{2k}]=\frac{(2k)!}{k!\cdot2^k}.$$ But I ...
3
votes
2answers
73 views

differentiation under the integral sign - moments

I want to find $$\mu_r' = \int_0^\infty y^r\theta e^{-\theta y}dy $$ A day ago I read something here on MSE about differentiation under the integral sign. I am not sure of how it works, however I ...
3
votes
1answer
168 views

Moment Generating function hard example!

X is a random variable with density $$f(x)=2e^{-2x+2} , x\geq1$$ and 0 otherwise. Determine $Mx(θ)$, the moment generating function for X, and the values of θ for which $Mx(θ)$ is defined. Use $Mx(θ)...
3
votes
2answers
6k views

How to find nth moment?

I'm quite new to the field so please bare with me. Problem: Let ξ be a random variable distributed according to a log-normal distribution with parameters μ and $σ^2$, i.e. log(ξ) is normally ...
3
votes
1answer
211 views

What's the difference between 'A sequence of moments' and 'moment generating function' when it comes to uniquely determine distribution function

I read from the MGB stats textbook which says something about "the problem of moments", as follows: "In general, a sequence of moments μ1,μ2..,μn,... does not determine a unique distribution function;....
3
votes
2answers
178 views

evaluating moment generating functions

Let $Z_1,Z_2,\ldots,Z_{14} $ be 14 independent N(0,1) variables, and let $Y=Z_1^2+Z_2^2+\cdots+Z_{14}^2$. Provide answers to the following to two decimal places. Evaluate the moment generating ...
3
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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_{...
3
votes
1answer
71 views

What is the origin of the term “moments” in the study of random variables?

I understand what the moments are, I just want to know who picked the term "moment" and why? How is the word "moment" related to different but related ways to describe the shape of a random variable?
3
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1answer
355 views

Using the mgf to get moments and the dominated convergence theorem

The moment generating function of $X$ is given by $M_X(t) = E[e^{tX}]$. I'm wondering if it is always possible to obtain moments as $E[X^k] = M_X^{(k)}(0)$, i.e. the $k^\text{th}$ derivative of the ...
3
votes
1answer
99 views

Show that the moment generating function of $ W$ is $M_W(t) = (qe^t+p)^n$

If $Y$ is a random variable with moment-generating function $M_Y(t)$ and if $W$ is given by $W=aY+b$, then the moment generating function of $W$ is $e^{tb}M_Y(at)$ Suppose that $Y$ is a binomial ...
3
votes
1answer
1k views

Computing variance of compound Poisson process from the M.G.F.

Let $X_t = \sum_{i=1}^{N_t} Y_i$ and $N_t$ be a Poisson process with intensity $\lambda >0$. Suppose $Y_i$ are i.i.d. (independent of $N_t$) with normal distribution $N(m,\sigma^2)$. Compute $...
3
votes
3answers
298 views

Finding moment generating function of $f(x)= \frac 1 {\theta^2} xe^{-x/\theta}$

I've been stuck on this question for a while now and my exam is coming up so,any hints/comments etc. would be greatly appreciated. Question: Find the moment generating function of the probability ...
3
votes
2answers
299 views

Statistics: Odd Moments

Need help with this stat question. I know you start by integrating $z^k f(z)$ from $-\infty$ to $0 +$ integral of $z^k f(z)$ from $0$ to $\infty$. After that I'm stuck.
3
votes
2answers
98 views

Boundedness of an integral of square function implying zero integral

Let $\alpha:\mathbb R\mapsto\mathbb R$ be the smooth function such that $$\int_{-\infty}^{\infty}[\alpha'(x)-x\alpha(x)]^2e^{-\frac{x^2}2}dx<\infty.$$ I wish to prove that $$\int_{-\infty}^{\infty}[...
3
votes
1answer
564 views

Sum of a random number of independent random variables

Consider the sum $Y = X_1 + \cdots + X_N$ where $N$ is a random variable that takes nonnegative integer values, and $X_1, X_2, \cdots$, are identically distributed random variables. Assume that $N, ...
3
votes
1answer
58 views

Confidence Interval for Chi-squared distribution

Let $X_1,\ldots,X_n$ be a random sample of size $n$ from a Beta distribution with parameters $α$ and $β=1,$ that the pdf is given by $f(x) = αx^{α-1}$ Find the distribution of $-2\alpha\sum\log x_i$ ...
3
votes
1answer
989 views

mixture distribution moment generating function

Let $X$ and $Y$ be independent random variables, with known moment generating functions $M_X(t)$ and $M_Y (t)$ and $I$ be such that $P(I = 1) = 1 − P(I = 0) = p \in (0, 1)$. Compute the moment ...