Questions tagged [moment-generating-functions]

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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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15 views

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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23 views

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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39 views

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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1answer
25 views

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 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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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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25 views

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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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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55 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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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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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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24 views

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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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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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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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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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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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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31 views

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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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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49 views

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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16 views

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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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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36 views

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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19 views

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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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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1answer
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moment generating function of the difference between two random variables

I need to find the moment generating function of $G = Y - X$ where $Y$ ~ exp($\frac{1}{2}$) and $X$ ~ exp(1). X and Y are independent. I read in another topic that $m_{Y-X}(r) = \frac{m_Y(r)}{m_X(r)}...
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173 views

Distribution of Dot-Product of Two Independent Multivariate Gaussian Vectors

Let $X,Y\stackrel{\text{i.i.d.}}{\sim}\mathcal{N}(0,I_d)$, where $I_d$ is the $d$-dimensional identity matrix. What is the distribution of $\langle X,Y\rangle=X^TY$? Approach 1: So far I know that ...
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Sums of trigonometric functions and polynomials

I have to calculate sums of the following forms $$\sum\limits_{k=1}^nP(k)f_m(kx),$$ where $P\in\mathbb{R}[X]$ and $f_m(x)=\sin^m(x)$ or $f_m(x)=\cos^m(x)$. This problem comes from consideration of ...
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Proof of Simple Facts about Moment Generating Functions

This is surely a very simple, well-known fact about moment-generating functions, though I am interested in the rigor required to prove it. It is surely the case that for any random variable, $X$, ...
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Finding a probability density from an exponential family using a moment generating function

I would like to find the density of a sum of i.i.d. random variables $\bar{Y}=\frac{1}{\nu}(Y_1+...+Y_\nu)$, where the density of these random variables is $f_Y(y;\theta)=e^{\theta y - b(\theta)}f_0(y)...
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1answer
34 views

moment generating function for $S_N=X_1+\cdots+X_N$ with $N$ dependent of $X_1$

Let $X_1, X_2, \ldots$ independent and identically distributed discrete random variables with support contained in $\mathbb{N}$ and let $N=X_1+1$. How can I calculate the moment generating function of ...
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Example iid variables $X_i$ where $S=\sum_{j=1}^NX_j$ but $M_S(t) \neq P_N(M_X(t))$

Let let $N$ a discrete random variable with support contained in $\mathbb{N}$. If for a fixed value of $N$ we have that $X_1, \ldots, X_N$ are independent and identically distributed random variables ...
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Is there any way to test a moment generating function_ [closed]

Like isnt there way to put the function value inside and then I would get the same value as the uniform probability function?
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How to find the MGF of this pmf?

A random variable $X$ has pmf $p(x;\alpha) = (1-\alpha)^{x-1} \alpha$ for $x = 1,2,\dots$. Find the moment generating function of $X$, $M_X(t)$. What I've done: $E[e^{tx}] = \sum_x e^{tx} (1-\alpha)^...
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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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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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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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130 views

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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128 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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141 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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43 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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255 views

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 ...