Questions tagged [moment-generating-functions]

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MGF of Normal Distribution

Let X $\backsim \mathcal{N}(0,\sigma^2)$ and $t>0$. I want to find $E(e^{-t x})$ $$E(e^{-t x})=\int_{-\infty}^{\infty} e^{-tx} \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{\frac{-x^2}{2\sigma^2}}dx = \int_{-\...
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Evaluating mean of uniform distribution using mgf

For the uniform [0,1] R.V, the mgf is $M(s)=\frac{e^s-1}{s}$ and the derivative is $M'(s)=\frac{se^s-e^s+1}{s^2}$ to calculate the mean we have to take the limit $\lim_{s\to0}M'(s)$. Why is it that ...
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the generated function of pulling a ball from a bin of balls with size that it is disturbuted by NP(2,p)

let's say that I got a bin of balls, which its size is distributed negative binomially: NP(2,p). And let's say that each of the balls is numbered from 1 to x-1. What is the generated function of Y: ...
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Rewriting a transform

From a question in my textbook Suppose that the transform associated with a discrete random variable X has the form $$M(s)=\frac{A(e^s)}{B(e^s)}$$ where $A(t)$ and $B(t)$ are polynomials of the ...
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Can a constant be a moment generating function of a random variable?

I need to disprove that there exists a random variable $X$ such that its MGF is $M_X(t)=e$. Not sure how to approach this...
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How to derive the Moment Generating Function for the Generalized Logistic Distribution? [duplicate]

I found the line $$ M_X(t)=E\{e^{tX}\}=\alpha\int_{-\infty}^\infty e^{-(1-t)x}(1+e^{-x})^{-(\alpha+1)}dx=\frac{\Gamma(1-t)\Gamma(\alpha + t)}{\Gamma(\alpha)} $$ in http://home.iitk.ac.in/~kundu/...
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Negative Binomial 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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Prove the Random Sample is Chi Square Distribution with Moment Generating Function.

$X_1$, $X_2$, and $X_3$ are random sample taken from normal distribution with $\mu=0$ and $\sigma^2=1$ (standard normal distribution). Let $Y=X_1^2+X_2^2+X_3^2$. Prove that $Y$ have distribution chi ...
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Opposite of 'If X,Y are independent, Mx+y(t)=Mx(t)My(t)'

It is known that If X,Y are independent random variables, Mx+y(t)=Mx(t)My(t) It's because if X and Y are independent, f(X) and g(Y) are also independent. How about the opposite? It X and Y have ...
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Use the MGF to derive all moments of $X \sim N(0, \sigma)^2$

Use the moment generating function to obtain the moments of all orders of $X$ if $X \sim N(0, \sigma^2)$ My attempt: The mgf of a generic $N(\mu, \sigma)^2$ is $M_X(t) = e^{\mu t}e^{(\sigma^2t^2)/2}$...
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Doubt regarding Moment generating function and Laplace transformation uniqueness.

Even though this is well know result I would like to deepen a little bit more in the conditions that allow us to derive this result. Suppose $\{X_i\}$ is a sequence of random variables, each with a $...
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Moment generating function of a gamma distribution

If I have a variable $X$ that has a gamma distribution with parameters $s$ and $\lambda$, what is its momment generating function. I know that it is $\int_0^\infty e^{tx}\frac{1}{\Gamma(s)}\lambda^sx^...
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Finding the convergence of moment generating function computed using integrals and series

Given the exponential density function $f_X(x)=\lambda e^{-\lambda x}$, the nth moment is $\mu_n=n!/\lambda^n$. If using the series representation to find the moment generating function $g(t)$: $$g(...
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Finding the moment generating function for independent trial with density $f_X(x)=\frac{e^{|x|}}{2}$

For an independent trial for the random variable X with density $f_X(x)=\frac{e^{|x|}}{2}$. If $S_n = X_1 + ... X_n$, $A_n = S_n/n$, and $S_n^*=\frac{S_n-n\mu}{\sqrt{n\sigma^2}}$, I found the ...
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Is the moment generating function of the gamma density $g(t)=(\frac{\lambda}{\lambda - t})^n$?

My book defines the gamma density as the following: $$f_X(x)=\lambda (\lambda x)^{n-1}e^{-\lambda x}/(n-1)!$$ And has the moment generating function of this density as $\frac{\lambda}{\lambda +t}$. Is ...
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Finding Variance from a joint moment generating function

The random vars X and Y have, for all real values of $T_1, T_2$, the joint mgf $M(T_1 , T_2) = \frac{1}{2} e^{T_1 +T_2} + \frac{1}{4} e^{2T_1 +T2} + \frac{1}{12}e^{T_2} + \frac{1}{6} e^{4T_1 +3T_2}$ ...
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uniform moment generating function at t=0

I have calculated the moment generating function for the uniform distribution as $$M_X(t)=\frac{e^{tb}-e^{ta}}{t(b-a)}$$ However I know $M_X(0)=1$ but I can't get my head around how this is possible ...
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51 views

Help with integration of first moment from PBE

I'm wondering anyone can help me with the following integration: $$\frac{d(m_0 V)}{dt} = BV$$ where $B$ is just a constant, $V$ is a variable parameter. Product rule must be applied somehow? EDIT: ...
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The expectation of log[1+e^(f)]

There are many examples about how to compute the expectation of $\log(1+e^x)$ such as approximating it with something like Maclaurin series. I have a slightly complicated situation \begin{equation} \...
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1answer
30 views

Moment Generating Functions Taylor series

So I'm revising moment generating functions and I'm stuck on a part of a question I'm looking at. So I am asked to find the moment generating function of a random variable X whose distribution is ...
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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 ...
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1k views

Show that X and -X are identically distributed and their moment, $M_{x}(t)$ is even

Let's assume X is a random variable with an even pdf. To show that X and -X are identically distributed, we need to prove that $F_X(x)=F_-X(x)$. We also know that X having an even pdf means $f_X(x)...
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57 views

computing the expectation of Trigonometric functions, “sin” and “cos”, with respect to their variables

How can I compute the expectation of $\cos((s\odot y)^T\alpha)$ with respect to all its variable $s$, $y$ and $\alpha$ where all of them are Gaussian distributions? $$\int \mathcal{N}(y|\mu_y,\Sigma_y)...
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1answer
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Deriving a probability from a moment-generating function

If the MGF of $X$ is $\beta^te^{t^2}$, for some $\beta>0$, find $\Bbb P (X>\text{log}_e(\beta))$. So far, I can see that the MGF of $X$ is similar to that of a standard normal distribution; ...
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1answer
22 views

Finding negative 2nd moment of gamma distribution

Given a gamma distribution with shape $\alpha=2$ and rate $\lambda=10$, I was first asked to find an expression for $\Bbb E[X^k] \ \forall \ k \in \Bbb N$. Directly computing this, I got $$\Bbb E[X^...
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How big are the exponential moments of a truncated normal distribution?

Given a random variable $X$ valued on $[-1,1]$ with mean zero. We can use say Hoeffding's Lemma to get $$ \mathbb E[e^{\lambda X}] \le e^{\lambda^2/2}$$ I believe this bound cannot be improved much ...
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Moment generating function of two variables

I am able to do all the parts except the very last. I have been trying to coax the differential equation $\frac{M'}{M}=t$ or something to that effect but I don't see how I can achieve this. Hints ...
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1answer
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Proof of if two random variables have the same distribution then they have the same moment generating function.

I am trying to prove that if $X$ and $Y$ have the same distribution, then they have the same moment generating function: $M_X(t) = M_Y(t)$ for all $t \in \mathbb{R}$. I came up with a proof, but am ...
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Central moment for a uniform distribution

The probability density function of T is given by $$f(t) = 1/2h \text{, for each } t\in(-h,h) $$ where $h > 0$. Derive an expression for the central moment I used integration and got $\frac{(b-u)...
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Find mean and variance from mgf where t is denominator

For continuous random variable X, pdf: $f_{X}(x)=2(1-x), x\in[0,1]$ mgf: $M_{X}(t)=\frac{2(e^t-t-1)}{t^2}$ Problem is to find mean and variance from mgf, I tried using $\frac{d}{dt}M_{X}(0)$ and $\...
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1answer
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Dominating function for derivative of moment generating function

Let $X$ be a random variable and the moment generating function $$\psi_X:(-\varepsilon,\varepsilon)\rightarrow \mathbb{R}_+,\quad \psi_X(t):=E[e^{tX}]$$ be defined, such that $\psi_X(t)<\infty$ ...
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How to transport a unique multimodal log-normal distribution by a set of moments?

I am using a CFD-QMOM approach to calculate the transport of nano particles through a fluid domain. The procedure that I am using is basically what is shown in this image: So I start with a certain ...
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Distribution of expectation operator when computing mgf of X bar

I'm trying to work through the proof for the moment generating function of $\overline{X}$. The proof below looks fairly straightforward but I'm having trouble understanding getting from the 2nd to ...
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Moment generating function (find the probability)

The moment generating function of a random variable $X$ is given by: $$M(t) = (1/3^{2k})(7+2e^t)^k, \quad \forall t$$ a) Determine $P(X = 3)$ b) Derive the $r^{th}$ factorial moment of $X$ I ...
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Calculating $\mathbb{P}(Y \leq 1)$ given the moment generating function

Given the moment generating function $$M_Y(t) =\frac{4-3t}{2(t-2)(t-1)}$$ with $t<1$ find $\mathbb{P}(Y \leq 1)$. First I tried to convert this to the probability generating function, because than ...
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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 ...
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1answer
52 views

Moment Generating Function of beta ( Hard )

Given $X$ is a random variable ~ $Beta ( a , b)$ distribution and $X$ belongs in (0,1) Does the (MGF ) $E[e^{tx}]$ exist for every value of $a , b$ ? (Mgf must not be equal to infinity in order ...
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Why does $\mathrm{E}[e^{-X}] = 0$ imply $\mathrm{P}(X = \infty)=1$?

Came across the following problem: For independent $(Y_n)_{n\geq1}$ with $Y_n \sim Exp(\lambda_n)$, let $X = \sum_{n\geq1} Y_n$. Show that if $\sum_{n\geq1} (\lambda_n)^{-1} = \infty$, then $\...
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Normal distribution non linear transformation

I have the following problem : Given $X \sim N(\mu,\sigma^2)$ and $X' = h(X) = (\frac{x-\mu}{\sigma})^2$ Find $E[X']$ and $V[X']$. My reasoning is as follow : Since $X' \sim (\frac{x-\mu}{\...
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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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the moment generating function of the negative binomial distribution

According to the textbook I use, it states that: $X$~$Neg.bin(x;k,\theta) = {n-1 \choose k-1}\theta^k(1-\theta)^{n-k}$ Which I have no problem. The problem arises when I try to find the moment ...
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24 views

factorial moment generating function

I'm trying to get the factorial moment-generating function of a binomial random variable. I know that $F_X(t) = E[t^x] = \Sigma_xt^xp(x)$ so I get $\Sigma_xt^x{n \choose x}\theta^x(1-\theta)^1-x$ ...
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Moment generating function of two Poisson distributions

The time between accidents on the Riverfront Bridge follows a Poisson process with a mean time of 40 days between accidents. The time between accidents on the Overview Bridge follows a Poisson ...
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differentiating(?) Poisson distribution

I've been facing this - i don't even know how to call it - problem for a few hours now and I have know idea how to "do" this. I mean... I feel like this has something to do with binomality of Poisson ...
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Example of higher random vector moments

While reading about random vectors, I learned that... $$ E\left[\vec{X}\right] = \left[\begin{array}{cccc} E\left[\vec{X}_1\right] & E\left[\vec{X}_2\right] & \cdots & E\left[\vec{X}_m\...
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456 views

Need help with moment generating function of geometric distribution

The cheat sheet I have tells me the moment generating function for Geometric Distribution is: $$M(t) = \frac{p}{1-(1-p)e^t} $$ But most resources and me personally working it out I get: $$M(t) = \...
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178 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 $X + 1 \sim \mathrm{geometric}(p)$ and determine $p$ in terms of γ. In order to solve this problem, I first ...
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Branching processes: Proof a limit of a probability

i have tried a question which the last part I couldn’t solve. Previously I have proved the explicit formula of $G_n={[n-(n-1)s]}/{[n+1-ns]}$ if that is useful Let Xn be the size of the nth generation ...
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How to use mgf to find the distribution of a standardised normal.

We are given that: $Z=\frac{Y-\mu}{\sigma}$ We want to show that, if $Y\sim N(\mu ,\sigma^2)$, then $Z$ is a standard normal random variable using the uniqueness of moment generating functions. The ...
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147 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 ...