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Questions tagged [moment-generating-functions]

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92 views

Calculating the conditional expected Value of correlated functions/ Moment Generating function

I have a little problem in a proof. I have to calculate the following conditional expectated value: $$ \mathbb{E} \left[ \varphi_{k}^{\Phi} C_{i,k-i}\vert \mathcal{T}_{t}, \Phi \right].\qquad \qquad (*...
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1answer
48 views

Moment generating function of sum of $N$ exponentially distributed random variables

We consider i.i.d. random variables $(X_i)_{i\geq 1}$, with $X_i\sim \text{Exp}(\lambda),$ and further independent variable $N \sim \text{Poisson}(\mu)$. Let $Y = \sum_{i=1}^N X_i$. Determine the ...
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0answers
194 views

Why the second cumulant is variance?

I have trouble understanding the term of second cumulant generating function. By the definition of cumulant generation function, it is defined by the logarithm of moment generating function $M_X(t)=E(...
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2answers
35 views

Why is this the result of computing the expectaion of this variable

I was given the following example with a solution: But I do not understand how they got from step 2 top step 3, i.e how they computed the result of the expectaion function.
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1answer
28 views

Methodology behind moment generating function $E(e^{t(A-B)})$ for two independent and identically distributed random variables A and B

For independent and identically distributed random variables A and B which both share a moment generating function (mgf) of $M(t) = E(e^{tA}) = \frac{3-t}{e^t}$. The mgf of A - B is, using methodology ...
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1answer
61 views

How to solve this integral involving exponential and exponential of exponential?

I would like to find the moment generating function of $$ e^{-a-bX^2}, $$ where X is $N(\mu,\sigma^2)$. This is equivalent to compute the following integral : $$ I = \int_{-\infty}^{+\infty}e^{te^{-...
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1answer
33 views

Moments of equally weighted mixture of two bivariate normal

Suppose I have two bivariate random vectors $\tilde{W}, \bar{W}$ distributed as follows $$ \bar{W}\sim \mathcal{N}(\begin{pmatrix}1\\-1 \end{pmatrix}, \begin{pmatrix} 1 & \rho_{\bar{W}}\\ \rho_{\...
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1answer
16 views

Independent exponential random variable

Let $X=X_1+...X_n$ where $X_i's$ are independent exponential random variables of parameter 1. (a) Show that the moment generating function of $X_1$ is $\frac{1}{1-t}$ (b) Compute the moment ...
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2answers
61 views

Help understanding moment generating functions

A moment generating function for a random variable $X$ is defined as: $M(t)=E(e^{tX})$ Now this is a nice and concise definition, but it's not descriptive at all, what role does $t$ play in this ...
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0answers
88 views

Identifying distribution of random variable given mgf

I have the following question, and am not too sure even where to start to solve this one! So any help or tips would be so much appreciated! “A random variable Y has moment generating function $M_Y(t) ...
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0answers
109 views

Relationship between autocorrelation function and characteristic function

Suppose $ X $ is a random variable. Then there is an associate characteristic function $\phi_X(t)=E(e^{itX})$. Then we know that the moments can be expressed as derivatives of $\phi_X$, $$EX=\frac{d\...
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1answer
208 views

Log of the Moment Generating Function

The Question: Let $X$ be a random variable such that $\Bbb E[X] = 0$ and $\text{Var}(X)=1$. Show that, as $t \rightarrow 0$, $$\ln M_X(t)= c t^2 + o(t^2)$$ where $M_X$ is the MGF of $X$, $o$ is the ...
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1answer
194 views

Intuition behind moments of random variables [duplicate]

I am looking to understand the intuition behind moments of random variables. I understand the first moment relating to the mean and the second one relating to the variance. But what use does the n-th ...
0
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1answer
25 views

plugging binomial moment function into poisson moment function

I have no idea how the author got from $$E[(pe^s+(1-p)e^t)^x]\rightarrow e^{\lambda(pe^s+(1-p)e^t-1)}$$ The author writes $a=e^t$, and then sets $a=pe^s+(1-p)e^t$ and plugs it in where $e^t$ ...
0
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2answers
75 views

conditional moment generating function author transforms solution

At the last step of the provided solution to this problem the author transforms $$\frac{e^{t(n+1)}-1}{e^t-1}=(1+e^t+e^{2t}+...+e^{nt})$$ It is probably something simple, but I am having trouble ...
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1answer
107 views

Derivation of MGF of a squared random variable X~N(0, 1)

Consider a continuous random variable $Y = X^2$ such that $X\sim N(0, 1)$. The PDF of this variable is $f_Y(x)= \dfrac{1}{\sqrt{2\pi}} x^{-\frac12}e^{-\frac{x}{2}}$ as discussed in this post. ...
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2answers
36 views

Alternating series of normal random variables and moment generating functions

Say $X_m$ is a normal distribution with mean $0$ and variance $2m$. Now suppose we have an alternating series $-X_1+X_2-X_3+...$ up to $X_n$. Find the distribution of this series. My working so far: ...
1
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1answer
32 views

Distribution of $Y=\Pi_{u=1}^n e^{X_i}$, where $(X_i)_i$ are i. i. d. with MGF $M_X(t)=e^{\frac{t^2}{2}}$

My working so far: $X_1=X_2=X_3=...=X_n$ are standard normal. $Y=e^{X_1}e^{X_2}...e^{X_n}=e^{nX}$ I have tried to find the distribution of $Y$ using this method:$$F_Y(y)=Pr(e^{nX}\le y)$$ $$=Pr(X\le ...
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1answer
17 views

Finding joint conditional expectation of binomial

In this question, the author finds the probability mass function and then finds the expectation because it happens to be a hypergeometric and therefore the answer is well known. In general, how do ...
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3answers
31 views

Cant understand skipped step in textbook solution

I can't understand this solution from my textbook. somehow $e^{tn}$ comes out of this expectation. But I don't understand why because inside the expectation it is $E(e^{(s-t)X_c}e^{tx}|x=n)$. I ...
0
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1answer
84 views

Finding distribution of $X$ from MGF

Suppose the moment generating function of $X$ is $$\sum_{n=1}^\infty \frac{6}{\pi^2}\frac{1}{n^2}e^{tn}\quad \text{for}\ t\le0$$ What is the distribution of $X$? We know that the MGF for a discrete $...
0
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1answer
41 views

Random sum of random variables with different variances

If $X_1, X_2,...,X_n$ are independent and have the same expected value, but differing variances, does Wald's equation (that is, $E[X_i]=E[N] E[X]$) still apply? In particular, my problem to find the ...
0
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1answer
51 views

where is $\sum_k(-1)^k $ going in this formula?

$\varphi _{X}(w)=\sum ^{\infty}_{k=0}(-1)^k(wb)^{2k}$,and the mth moment of X is $E[X^m]=(-j)^m\frac{d^m \varphi _{X}(w)}{dw^m}=(-j)^m\sum_k (-1)^k \times 2k \times(2k-1)\times ... \times(2k-(m-1))\...
0
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1answer
64 views

Even moments = 0 iff $X=0$?

Kind of a dumb question. I wasn't able to find this in probability texts, either elementary (Larsen and Marx) or advanced (David Williams), so it's probably wrong. Is this false? Let $X$ be a ...
0
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0answers
89 views

Do the derivatives of a moment generating function evaluated at 0 uniquely determine the distribution of a random variable?

I understand that the moment generating function of a random variable uniquely determines that random variable. And that for an rv X, M(0)=1, M'(0)=E[X], M''(0)=E[X^2],... and so on. My question is ...
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0answers
29 views

Given kth moment determine the distribution

Suppose that $Y$ is a random variable such that $E(Y^k) = \frac14+2^{k-1}$ $k = 1, 2, . . .$ Determine the distribution of $Y$. As I can recall for probability generating function $g_y^{k}(1)=E(Y^k)...
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1answer
23 views

moment generating function vs density function of sum of two uniform rv's

Suppose we have two random variables $A$ and $B$ both uniformly distributed on $[0, 1]$. Define $Z=A+B$. The moment generating function of $A$ and $B$ is $M_A(\theta)=M_B(\theta)=\frac{e^{\theta}-1}{\...
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0answers
33 views

calculating probability distribution function of double integral moment

When k th moment is as followed, what would be the probability distribution function of X? (k=0, 1, 2...) $$E[X^k]=(2k+1)!!$$ I thought about central normal distribution's moment, but couldn't think ...
3
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2answers
38 views

Show that $\varphi_X(2\pi k)=\operatorname E(e^{i2\pi kX})=1 $

Let $X$ be a random variable and $X \in\mathbb Z.$ Show that $$\varphi_X(2\pi k) = \operatorname E\left(e^{i2\pi kX}\right)=1 $$ for $k \in\mathbb Z$ I tried to expand the expected value by its ...
4
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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 $$ \...
1
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2answers
238 views

Expected value Taylor series

If $X ∼ N(0, σ^2)$, use the moment generating function of X to show that $E(X^r)= 0$ for r odd, and $E(X^{2r} ) = \frac{(2r)!σ^{2r}}{2^r(r!)}$ for $r=0,1,2,3.....$ Attempt: $m_X(u) = E(e^{Xu}) = E(...
0
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1answer
74 views

Deriving the joint MGF for $U = XY + (1-X)Z$, $X \sim \text{Bernoulli}(p=1/3), Y \sim \text{Exp}(\theta = 2), Z \sim \text{Poi}(\lambda = 3)$

The question is to find the MGF of $U = XY + (1-X)Z$, where: $X \sim \operatorname{Bernoulli} (p = \frac{1}{3})$ $Y \sim \operatorname{EXP} (\theta = 2)$ (Note that the notation is $\theta = \...
1
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0answers
47 views

Distribution of Product of Arbitrary Number of Random Variables

Suppose $X_i\sim f_{X_i}(x)$, with each $X_i$ independent and identically distributed for some arbitrary probability density function $f_{X_i}(x)$. Given this information, and this information only, ...
0
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1answer
91 views

Unique solution to an integral equation? (Moment generating function)

I have a solution to the following integral equation: \begin{equation} \gamma(t) = \dfrac{2}{t}\int_{0}^{1} \gamma(t x)(e^{(1-x)t}-1)dx, \, \gamma(0)=1. \end{equation} Can one show that the ...
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1answer
620 views

Relationship between Laplace transform and moment generating function for queue

(Again this is based on pp240 - 242 of the 1966 edition of Cox and Miller's "The Theory of Stochastic Processes"). So we have, for a queue in equilibrium/stationary a probability density function for ...
0
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2answers
31 views

moment generating function basics

if X,Y discrete random variables such that Y=aX+b , how do I prove $M_y(t)=e^{tb}M_X(at)$. My attempt: $M_y(t)$=$\sum$$e^{tax}.e^{tb}p(y)$. but then I notice this works iff p(x)=p(y) why is this ...
2
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0answers
58 views

Recursive Relation to obtain a MGF

Main Question : In an article I was reading there is a recursive relation suggested to obtain the moment generating function of a random variable by setting: \begin{align*} M_1(s) &= \frac{a_1}{s-...
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1answer
242 views

Bounded Pareto Distribution Moment Generating Function

I am interested in the Bounded Pareto Distribution but I can't find any reference that supplies me the moment generating function. This does however seem like something that is known. More general : ...
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1answer
48 views

Moment Generating Function under a new probability measure [closed]

Let Z be a standard normal random variable under a probability measure P. Set X = exp(θZ - θ2/2). Define a new probability measure P1(A) = E(X 1A), where the expected value is taken under ...
0
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1answer
77 views

A combinatoric solution (closed expression) for $\sum_{k=i}^n \binom{n}{k}p^k(1-p)^{n-k}$

I am trying to find a combinatoric solution for $\sum_{k=i}^n \binom{n}{k}p^k(1-p)^{n-k}$. i.e. write it as a closed function (and not as a sum). I know that in generating functions one can for ...
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0answers
70 views

Suggestion for probability problem book

I am an undergraduate having a course on probability that currently encompasses the topics: General Theory of Expectation, Modes of Convergence (almost surely/conv in probability), Laws of Large ...
1
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1answer
3k views

Proof of the Central Limit Theorem using moment generating functions

Below is a method of proving the Central Limit Theorem using moment generating functions. Let $$X_{1},X_{2},...,X_{n}$$ be a sequence of i.i.d. random variables with expected value and variance $$E(...
3
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2answers
164 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 ...
0
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0answers
34 views

Can we use uncorrelated to prove independent?

We have learnt in the course that the independence of two variables implies they are uncorrelated. i.e. $$p(X,Y)=p(X)p(Y)\Rightarrow E[XY]=E[X]E[Y]$$ We do know that the reverse doesn't hold. But can ...
1
vote
1answer
425 views

Determine pdf from mgf

If I am given a moment generating function $M(t)$, such that $$M(t) = \frac14 + \frac34\cdot\frac1{1-t}$$ where $t>1$, how can I know its probability density function?
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1answer
488 views

Is the sample mean of a Poisson Distribution Normal? [closed]

I am trying to figure out why we are allowed to use the sample mean in a normal distribution. I tried using moment generating function for $\bar{x}$ to prove that the distribution was normal but ...
0
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1answer
55 views

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
vote
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 ...
1
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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 ...
1
vote
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 ...