Questions tagged [moment-generating-functions]

Description added to tag

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A proof regarding the moment generating function.

We are required to prove that two random variables $(X,Y)$ are independent if and only if $m_{X,Y}(t_1,t_2)=m_X(t_1)m_Y(t_2)$ where $m(\_)$ is the moment generating function. Supposing $(X,Y)$ are ...
1
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1answer
36 views

Showing for a variable $X$ with mean and variance 1 that $\mathbb{E}\left(e^{X}\right)\geq e$

Let $X$ be a discrete non-negative variable with $\mathbb{E}\left(X\right)=Var\left(X\right)=1$. I'd like to show that necessarily $\mathbb{E}\left(e^{X}\right)\geq e$. My initial intuition was to ...
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0answers
52 views

Stochastic Geometry : Obtaining an Integral

I was reading some paper on stochastic geometry when i came across this integral. I was trying to verify results myself but this one seems to be disappointing. Although i do the instructions ...
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1answer
107 views

Mathematical Statistics - Jun Shao - 2.12

I'm trying to solve the following problem: Show that the m.g.f. of the gamma distribution $\Gamma(\alpha,\gamma)$ is $(1-\gamma t)^{-\alpha}$, $t<\gamma^{-1}$, using Theorem 2.1(ii). This ...
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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 $-2\alpha\sum\log x_i$ ...
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1answer
60 views

Proof verification : Inequality involving moment generating function (MGF)

The problem is as follows. My solution : Let $t \geq 0$ be arbitrary. Then $E(e^{tX})=\int_{-\infty}^{\infty} e^{tx}dF(x) \geq \int_{0}^{\infty} e^{tx}dF(x) \geq \int_{0}^{\infty} dF(x)=P\{X \geq 0\}...
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1answer
53 views

Derivation of an inequality regarding absolute moments and MGF

Let $X$ be a r.v. with MGF $M(t)$, which exists for $t \in (-t_0,t_0)$, where $t_0>0$. To show that $$E|X|^n < n!s^{-n}[M(s)+M(-s)]$$ I've reduced the problem to showing that $$2\{1+\frac{s^2}{...
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1answer
107 views

Computing $\iint_{\mathbb R^2}\exp\left(u\frac{xy}{\sqrt{x^2+y^2}}+v\frac{x^2-y^2}{2\sqrt{x^2+y^2}}-\frac12(x^2+y^2)\right)dxdy$

I'm trying to work on another solution to this question by computing the moment generating function: $$M(u,v)=\iint_{\mathbb{R}^2}\frac1{2\pi}\exp\left(u\frac{xy}{\sqrt{x^2+y^2}}+v\frac{x^2-y^2}{2\...
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0answers
139 views

Upper bound on P(S_n > a) by mgf, Strong Law Large Numbers Proof Casella, Berger

Having trouble coming up with a proof for a problem related to SLLN using MGFS. I found Strong Law of Numbers for $S_{n}$ Bounded Casella Berger 5.38 but the answer doesn't seem to hint at how we ...
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0answers
90 views

How are skewness, kurtosis etc. distributed?

The mean of independent identically distributed random variables, if the moments exist, is normally distributed (to second order). The variance of independent identically distributed random variables ...
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1answer
36 views

Invertibility of Moment Generating Functions Equal on Small Interval

I am taking a statistics course right now and we're using Hogg, McKean and Craig's Introduction to Mathematical Statistics (2013). How do you prove Thm 1.9.1: Let $X$ and $Y$ be random variables ...
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1answer
381 views

How to find the moment generating function of a poisson translation.

I want to find the mgf of $Y$, where $Y = ((X − λ)/√λ)$ and $X$ ~ Poisson$(λ)$. What steps should I take in order to arrive to the mgf of $Y$?
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46 views

Improper integral and moment-generating function of a function of exponential distributions

So I was trying to find the moment-generating function to the following random variable: $X^{2}+Y^{2}$ such that $X\sim Exp(\lambda)$ and $Y\sim Exp(2\lambda)$, independent to each other. I thought of ...
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1answer
351 views

Beta Distribution Moment Generating Function

I've seen that the moment generating function of the Beta Distribution is the following: $${\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\...
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0answers
13 views

Higher order moment estimation using the information of low order moments

Let $x \in \mathbb{R}$ be a random variable. Given the finite sequence of the moments, e.g., $E[x^{\alpha}], \alpha=0,..., N$, how we can obtain the higher order moments e.g., $E[x^{\alpha}], \alpha \...
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1answer
185 views

Joint moment generating function of $X_i$'s where $X_i$ is a linear combination of standard normals

I have a collection of $Z_1, ... Z_n$ which are all i.i.d standard normals. I have a collection of $X_1, ..., X_m$ such that each $X_i$ is a (potentially distinct) linear combination of $Z_1, ..., Z_n$...
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1answer
57 views

Given $E(X^r)$ find MGF and PMF

If $E(X^r) = 5^r, r = 1,2,3, \ldots $, how would you find the moment generating function of $X$ and the PMF of $X$? So far I have$$ M(t) = M(0) + \sum_{i=0}^\infty 5^r \frac{t^r}{r!}.$$ Am I doing it ...
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1answer
122 views

Generate random numbers using moment information

Let $x\in \mathbb{R}$ be a random variable. Given the finite number of moments of random variable $x$, e.g., $E[x^{\alpha}], \alpha = 0,...,N$, how we can generate random numbers with the distribution ...
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2answers
2k views

Find Moment Generating Function from Probability Mass Function

I need help understanding how to find the MGF using a PMF. The PMF is $f(x) = \frac{1}{2^{x-1}}$ when the random variable $X \geq 2$. I get that you need to multiply $e^{tx}$ by $\frac{1}{2^{x-1}}$. ...
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1answer
482 views

Chi-square with $n$ degrees of freedom,Normal distribution

How can I see that the $\chi^2(n)$ random variable has the moment generating function $$(1-2t)^{-n/2}$$? I would also like to know why precisely that if $Z \sim N(0,1)$ then $Z^2 \sim \chi^2(1)$. ...
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1answer
44 views

inequality with incomplet gamma function ( weibull distribution conditional probability)

i'd like to prove the following inequality: $$ (\int_{0}^{\infty}e^{-\alpha ((A+t)^b-A^b)} dt )^2 \geq \int_{0}^{\infty}t e^{-\alpha ((A+t)^b-A^b)} dt $$ where $\alpha \geq 0$ (scale parameter), $b \...
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0answers
75 views

Shifted log normal power expectancy

I want to come back on an old post here. Can someone compute $E(Y^{\alpha})$ for $\alpha \in \mathbb{R}$ where $Y$ follows a shifted log normal distribution? I saw here for the case where $\alpha$ ...
2
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1answer
344 views

Chi-Squared 2df = 2nλ/λˆ MGF Needed?

Question: Assume that $T_i (i = 1, ..., n)$ are independently identically distributed with exponential distribution with hazard $\lambda$. Assuming there is no censoring, prove that: $$\frac{2nλ}{\...
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1answer
141 views

Sums, maximums, and minimums of exponential random variables

Let $y_1$, $y_2$,...,$y_{10}$ be independent exponential random variables with mean $1$. (a) Find the distribution of their sum. (b) Suppose the random variables above represent the lifetime (in ...
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1answer
167 views

Finding the moment-generating function $f(x)=2*(\frac{1}{3})^x$

For this problem, I am tasked with finding the moment generating function of $X$ given it has the following discrete probability distribution: $$f(x)=2*(\frac{1}{3})^x, x\in\mathbb{N}$$ By the ...
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1answer
33 views

Moment generating function of $f(x)=\frac1{2^x}$

So I'm having trouble at figuring out how to calculate the MGF of the geometric function for the probability density function of $f(x)=\frac1{2^x}$ where x is discrete. I know that $$M(t)=\sum_{x=1} ...
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1answer
1k views

Conditional Moment Generating Function and Iterated Rule

Let $N$ by the number of claims made by a person. Assume that the Number of claims varies with the type of person. Measure the type of person by a second random variable $\Lambda$, which is ...
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1answer
554 views

Joint Moment Generating Function from Conditional and Marginal Distribution

Suppose that that random variable $N$ follows a Poisson distribution with mean $\lambda=6$. Suppose that the conditional distribution of the random variable $X$, given that $N=n$, follows that of a $...
2
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3answers
143 views

Let $X$ and $Y$ be independent and identically distributed random variables with moment generating function then $E(\dfrac{e^{tX}}{e^{tY}})$

Let $X$ and $Y$ be independent and identically distributed random variables with moment generating function $M(t)=E(e^{tX});\ \ -\infty<t<\infty$ then $E(\dfrac{e^{tX}}{e^{tY}})$ equals ? $(A)=...
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1answer
29 views

Is $\mathcal{L}_{M}(\Omega, \mathcal{F}, \mathbb{P})$ a linear subspace?

Is the space $\mathcal{L}_{M}(\Omega, \mathcal{F}, \mathbb{P})$ of of all random variables, $X$, whose moment-generating function $\mathbb{E}[e^{tX}]$ exists for all $t \in \mathbb{R}$ a linear ...
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0answers
234 views

How to prove sample variance has a gamma distribution by using mgf.

If $X_1, X_2, ..., X_n \sim \mathcal N (\mu, \sigma^2) $ i.i.d. and $S^2$ is the sample variance. How do I go about calculating the moment generating function of $S^2 $ itself and deduce that $S^2$ ...
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1answer
501 views

Finding a PDF from a MGF

I've got a problem from a STEP III paper which I am trying to answer. I have got as far as $$M_T (\theta) = \frac{1}{(1-\theta)^2}$$ and $$ \int_{0}^{\infty} g(u) e^{\theta u} \mathrm{d}u = \frac{...
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1answer
168 views

Variance Derivation of Chi-Squared Distribution

If we find the MGF of $$Z^2$$ where $$Z \sim N(0,1)$$ we find that $$M_{Z^2}(t) = (1-2t)^{-.5}$$ for $t < .5.$ Consider that a chi squared distributed variable $X$ with $r$ degrees of freedom is ...
2
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1answer
219 views

Moment Generating Function from Piecewise Constant CDF?

I am trying to find the moment generating function from a piecewise CDF that has constant values: $$F(x)=\begin{cases} 0, & \text{if } x<0, \\ \frac14, & \text{if } 0\le x<2,\\ \frac34,...
2
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1answer
380 views

moment-generating function for uniform discrete distribution

$a)$ Consider an uniform discrete distribution ( $X$ ~ uniform{$0,...,n-1$} ) and find the moment-generating function. $b)$ Now Consider $Y$ ~ uniform{$\frac{0}{n},...,\frac{n-1}{n}$} and find the ...
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1answer
68 views

Moment Generating Function to Distribution

I am trying to find the distriubtion of X when $M_X(t)=\frac 16 e^t+ \frac{2}6e^{2t}+\frac{3}6e^{3t}$ With some simple computations, I found that $Var(x)=5/9$, and $E(x)= 7/3$ However, since the ...
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3answers
102 views

Is it possible to find an upper bound on the moment generating function of $\sqrt{|X|}$, where $X\sim \mathcal{N}(0,1)$?

So basically we need to find an upper bound on the integral $$\int_0^\infty \exp(f(x)) dx,$$ where $f(x) = \lambda \sqrt{x} - x^2/2$ with $\lambda>0$. I used the Laplace's method of approximation ...
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How do I find the moment generating function of $N(0, 1)$ [closed]

Here is my homework, I just don't really know how to find the moment generating function. I can do the rest after i get that. Any help?
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1answer
345 views

With K(t) = log(E[e^tX]) ,show that K'(0) = E[X]

Just as the title, it is Question 83 on page 90(Ross's book, Introduction to Probability Models) since K(t) = $log(E[e^{tX}])$ Convert $log_{10}$ to $ln_e$, K(t) = $ln(E[e^{tX}])$/$ln$10 Then K'(t)...
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0answers
49 views

Problem with mixture of distributions solution

I have a problem with two competing solutions giving different answers. I know the right one, but I want to understand why the other solution does not work. (Paraphrase from an ACTEX SOA Exam P ...
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1answer
37 views

Solving $\sum_{k=n}^{\infty}e^{tk}q^{n}(1-q)^{k-n}\binom{k-1}{n-1}$

I am trying to find the moment generating function of a variable with a negative binomial disribution (Counuting the number of trials when the counting stops once $n$ successes, with probability $q$ ...
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0answers
281 views

Does Chi-square distribution follow a Standard Normal distribution under Central Limit theorem? If yes, how do I prove it?

I tried using the Moment generating function technique..The Mgf is (1-2t)^(-n/2) for a chi-square distribution with n degrees of freedom..Now how do I proceed and converge this Mgf to that of a N (0,1)...
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1answer
158 views

Evaluating an $n$-fold convolution

Let $f(y) = e^{y-e^y}$. I want to find $$ f_{(n)} = \underbrace{f * f * \dots * f}_{n} $$ where $*$ denotes convolution. I'm sure that Fourier or Laplace transforms are the key but I don't have a lot ...
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1answer
45 views

Alternative method to find maximum point

I'm trying to find the point where the maximum of my function occurs. Because the function does not yield trivial expressions of derivatives, I can't just differentiate and find the root. I'm trying ...
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1answer
140 views

Finding the mgf of $f_X(x) =e^{−2|x|}I_{(−\infty,\infty)}(x)$

Give the mgf of a random variable having pdf $$f_X(x) =e^{−2|x|}I_{(−\infty,\infty)}(x)$$ I found a very similar problem here but I wanted to make sure I'm applying it correctly: $$ \begin{align} ...
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0answers
442 views

Finding mgf of a gamma random variable and finding $3rd$ moment

Give the mgf of a gamma random variable having mean $6$ and variance $12$, and then use the mgf to obtain the $3rd$ moment of the random variable. I have that \begin{align} \operatorname{Var}(X)&...
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0answers
37 views

How can I find this moment generating function?

I have a random variable $C$ with probability density $\frac{c}{8}$ where $0\leq c \leq 4$. I am having trouble finding the moment generating function of this distribution. $\begin{split} M_C(t) &...
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0answers
31 views

Expectation of standard deviation

The moment generating function of a gamma random variable is $M_X(t)=(1-t/\beta)^{-\alpha}$, we have $E[X]=\alpha/\beta$ $E[X^2]=\frac{\alpha(\alpha+1)}{\beta^2}$ $E[X^3]=\frac{\alpha(\alpha+1)(\...
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1answer
129 views

Ordinary and Moment Generating Functions

Let $p$ be a probability distribution on $(0,1,2)$ with moments $\mu_1=1$ and $\mu_2=\frac{3}{2}$ Find its ordinary function $h(z)$. Using this then find its Moment generating function. Then find its ...
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1answer
1k views

Use the convolution formula to find the pdf

Let $X$ and $Y$ be two independent uniform random variables such that $X\sim unif(0,1)$ and $Y\sim unif(0,1)$. A) Using the convolution formula, find the pdf $f_Z(z)$ of the random variable $Z = X + ...