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Questions tagged [moment-generating-functions]

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2
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24 views

How big are the exponential moments of a truncated normal distribution?

Given a random variable $X$ valued on $[-1,1]$ say we can use Hoeffding's Lemma to get $$ \mathbb E[e^{\lambda X}] \le e^{\lambda^2/2}$$ I believe this bound cannot be improved much if for example $X$...
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19 views

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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1answer
12 views

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 ...
2
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1answer
26 views

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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1answer
10 views

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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29 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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1answer
24 views

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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1answer
47 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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23 views

Integral Is not finite?

How can we prove with calculations that the below integral is not finite ` for $t\ge$ $\dfrac{1}{2}$ $\int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}}e^{-x^2(1/2-t)}\, dx.$ I would be pleased if you ...
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2answers
48 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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2answers
14 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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9 views

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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1answer
17 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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1answer
59 views

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 ...
0
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1answer
30 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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6 views

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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16 views

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
12 views

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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1answer
83 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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0answers
18 views

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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0answers
19 views

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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0answers
41 views

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)...
3
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1answer
33 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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1answer
22 views

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 ...
1
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1answer
21 views

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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2answers
18 views

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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0answers
25 views

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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13 views

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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29 views

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\...
0
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1answer
35 views

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 $$ E[(X -EX)^s] = \left. \frac{\partial \ln{M}}{\partial t^s} \right|_{t=0} $$...
3
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0answers
107 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 ...
4
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2answers
111 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 ...
0
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1answer
103 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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1answer
40 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 $...
2
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2answers
195 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 ...
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1answer
29 views

Ideas of how to find the distribution of Gamma(N,$\lambda$), with N~Geom($\alpha$)

I'm not exactly sure how to go about this, could I use the definition of the MGF or something along those lines?
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1answer
93 views

Sub-Gaussian and “nearly” sub-Gaussian random variables

Define $\Psi_{X}(\lambda) = \log E e^{\lambda X}$, and suppose that $EX=0$. We say that the random variable $X$ is sub-Gaussian with variance factor $v$ if: $$\Psi_{X}(\lambda) \leq v\lambda^{2}/2$$ ...
0
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1answer
35 views

method of moments and maximum likelihood estimators

I'm looking to find the estimate of $\mu$ for $n$ data using the method of moments and the maximum likelihood for the pdf given by $f(x) = \begin{cases} e^{-(x-\mu)}, & \text{if} \, x \geq \mu \\ ...
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1answer
36 views

Is the moment generating function smooth by definition?

So I'm going through Casella's Statistical Inference, and in Definition 2.3.6 he defines the moment generating function of a random variable $X$ with cdf $F_X$, denoted by $M_X(t)$, as $$M_X(t) = Ee^{...
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0answers
24 views

Method of moments when the first moment is $0$

I have a quick question regarding the method of moments estimator. Generally, when you have $k$ parameters you want to estimate, it suffices to find $k$ equations using $k$ moments. If you are ...
1
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1answer
23 views

What is the distribution of $P_M(M_B(t))$

$M_X(t)= P_M(M_B(t))$ $P_M(s)= (1-q+qs)^2 $ $M_B(t)= \frac{\beta}{\beta-t}$ Where P(x) is the probability generating function and M(x) is a moment generating function I identified M as $M \sim ...
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0answers
27 views

Moment Generating Function exercises: Knowing that $M_{X}(0)=1$ and $M´_{X}(0)= EX$

Hi guys, Any help with letter b of this exercise from Casella´s Book? I could finish the letter a). But I cant move in letter b). Any help? Thanks
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1answer
64 views

Assume that $Z_i\sim\mathcal{N}(0,1)$ are independent and prove that $\sum\limits_{i=1}^{n}(Z_i-\overline{Z})^{2}\sim\chi^{2}_{(n)}$

Assume that $Z_{i}\sim\mathcal{N}(0,1)$ are independent and prove the following results (a) $\displaystyle\overline{Z} = \frac{1}{n}\sum_{i=1}^{n}Z_{i}\sim\mathcal{N}(0,1/n)$ (b) $\overline{Z}$ and $...
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0answers
47 views

Moment generating functions and normal distributions

(a) If $X_{i}\sim\mathcal{N}(\mu_{i},\sigma^{2}_{i})$ are independent and $a_{i}\in\mathbb{R}$, $1\leq i \leq n$, then \begin{align*} Y = \sum_{i=1}^{n}a_{i}X_{i}\sim\mathcal{N}\left(\sum_{i=1}^{n}a_{...
2
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0answers
74 views

Given independent random variables $X_{k}\sim\text{Poisson}(\lambda_{k})$, what is the distribution of $X_{1} + X_{2} + \ldots + X_{n}$?

Show that, if $X_{k}\sim\text{Poisson}(\lambda_{k})$ and they are independent for $1 \leq k \leq n$, then \begin{align*} Y = \displaystyle\sum_{k=1}^{n}X_{k}\sim\text{Poisson}\left(\sum_{k=1}^{n}\...
1
vote
1answer
108 views

What is the moment generating function of the Laplace distribution?

Consider a random variable $X$ whose probability density function is given by $$f_{X}(x) = \frac{\lambda}{2}\exp(-\lambda|x-\mu|)\quad\text{for}\quad x\in\textbf{R}, \lambda > 0,\,\,\text{and}\,\,\...
1
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0answers
42 views

Calculate Probability from Moment Generating Function

Let 𝑋 and 𝑌 be two discrete random variables with the joint moment generating function $$𝑀_{𝑋,𝑌}(t_{1},t_{2})=(\frac{1}{3} e^{t_{1}} + \frac{2}{3})^{2} (\frac{2}{3} e^{t_{2}} + \frac{1}{3})^{3}...
1
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1answer
155 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 ...