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Questions tagged [moment-generating-functions]

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695 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ $$\...
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599 views

Uniform convergence of Empirical Moment Generating Function

In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as: $$ \begin{...
5
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407 views

Moment generation function -> characteristic function uniqueness

Here's my proof that moment generation function (if exists) uniquely determines characteristic function. Can you please see how to make it more rigorous or improve in either way (e.g. by citing ...
3
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116 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 ...
3
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Standard Normal Moments and Combinatorics

At around 16-17 mins in this video, the professor calculates the even moments of the standard normal. If $Z \thicksim N(0,1)$ then $$\mathbb{E}[Z^{2n}] = \frac{(2n)!}{2^n \cdot n!}.$$ The right hand ...
3
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65 views

Find the first moment of a probability distribution governed by a nonlinear first order ODE

May I ask if there is any standard way to find the first moment of a probability distribution governed by a nonlinear first-order ODE. For example, $$ \frac{\mathrm d p(x)}{\mathrm dx} = \alpha(x) p(x)...
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89 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 ...
3
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29 views

Asymptotics of moment

Suppose that I am given a probability distribution function on some Euclidean space. The question is when something can be said about asymptotics of the moments (for the order of the moment tending to ...
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887 views

Moment Generating Function for $r$th central moment

When using moment generating functions, to find the $n$th raw moment ("$n$th moment about the origin"), you take the $n$th derivative of the MGF and evaluate at $t=0$. To find the $m$th central ...
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38 views

Changing domain of the CGF of a stochastic process

Given a stochastic cadlag process $(X_t)_{t\geq 0}$. Define $A_{t}$ as the domain of the cumulant generating function by $f_t(s):=\log E(e^{sX_t})$, for which this expression is welldefined. I search ...
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629 views

Moment Generating Function for Brownian motion's exit of interval.

Let $B(t)$ be a standard BM. Consider the stopping time $T = \inf\{ t > 0: |B(t)| = a\},$ the usual first exit time of the interval $(-a, a).$ We can see that $\mathbb{E} e^{tT} < \infty$ for $...
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450 views

Compute conditional expectation from moment generating function

Let $X$ and $Y$ be real-valued non-independent random variables and suppose I have their joint moment generating function $$\phi(s,t)=\mathsf{E}\left[e^{sX+tY}\right],$$ which I assume is a "nice" ...
3
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384 views

Moment-generating function of a generalised normal random variable

Let $X$ be a random variable that follows the "version 1" generalised normal distribution described here, with p.d.f. $$f_X(x;\mu,\alpha,\beta)=k\exp\left\{-\left(\frac{|x-\mu|}{\alpha}\right)^\beta\...
3
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402 views

Normal Approximation to Binomial Distribution using Moment Generating Functions

We're told not to use the central limit theorem to show that the normal approximation is suitable for a binomial distribution when n tends to infinity. I've managed to show the answer, but it ...
3
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778 views

Why the probability characteristic function is always exist but moment generation function is not always exist?

I know that the characteristic function is always exist and moment generation function is not always exist instantly but don't know exactly mathematically.
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41 views

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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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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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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76 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}\...
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66 views

Using MGF's to characterize a distribution

Let $X_1,X_2,X_3$ be independent such that for all $x > 0$, $$ P(|X_i| > x) < e^{-x}, \;\;\; i = 1,2,3 $$ Prove that if $X_1+X_3$ and $X_2+X_3$ have the same distribution, then so ...
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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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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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290 views

Moment generating function of $Z=X-Y$ where $X,Y$ are independent binomial RVs

$X$ and $Y$ are statistically independent random variables. $X\sim \text{Bin}(n_1,p_1)$ $Y\sim \text{Bin}(n_2,p_2)$ $Z = X -Y$ $q_1 = 1 - p_1$ $q_2 = 1 - p_2$ I am trying to find the moment-...
2
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284 views

MOM estimator same as MLE estimator for normal distribution

Q: Let X1,...,Xn be IID N(μ, σ²) r.v. Is the MOM estimators for σ² and μ same as the MLE estimators? For Uniform distribution, i remembered that the MOM and MLE estimators corresponds and are the ...
2
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166 views

Properties of moment generating functions

Let $X$ be a random variable with density $f(x)=2x\cdot I_{0\le x\le 1}$, a cdf $F(x)=I_{\{x>1\}}+x^2I_{\{0\le x\le 1\}}$ and let $F_I(x):=1/\mu\int_0^x (1-F(y))dy$ where $\mu=E[X]$ (Note: $F_I$ ...
2
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67 views

Probability distribution whose moments are 2n choose n

The probability distribution whose odd moments are zero and whose even moments are the catalan numbers is the semicircle distribution. What is the distribution whose even moments are n choose n/2? ...
2
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137 views

Moment-generating function

Let X~Exponential(1). I know that the moment-generating function of an exponential distribution is defined as $(1-t\lambda)^{-1}$. And hence $E[e^{tx}]=(1-t)^{-1}$. But what is $E[Xe^{tx}]$? Would ...
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167 views

Wald martingale - applications?

I'm starting the study of stochastic processes. What are some of the applications of the Wald martingale? By Wald martingale I mean the following (defined by my professor). Let $\{X_n\}_{n\in\...
2
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115 views

Functions for which all moments are well-defined and finite

I understand Schwartz functions (informally) as the set of functions $f$ such that $f \in C^\infty$, $f$ decays at $\pm \infty$ faster than any polynomial, $\frac{d^nf}{dx^n}$ decays at $\pm \infty$ ...
2
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416 views

Moment Generating Function $X^2$ and $XY$

Let X and Y be two independent standard normal random variables. (a) Find the moment generating function of $X^2$. (b) Find the moment generating function of $XY$ . (c) Prove or disprove that $X^2$ ...
2
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230 views

Suppose a random variable X has the moment-generating function

$M_X(t)=\frac{1}{7}e^{2t}+\frac{3}{7}e^{3t}+\frac{2}{7}e^{5t}+\frac{1}{7}e^{8t}$ $t\in{\rm I\!R}$ a)Convince yourself that X has discrete distribution. What is the pmf of X? b)Find the 3rd moment of X....
2
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0answers
116 views

Question concerning Lundberg-Exponent

My question is based on the beginning of Chapter 8.3.2 in the book "Modelling Extremal Events" by Embrechts,Klüppelberg and Mikosch. We consider a Cramer-Lundberg-Model and assume that the conditions ...
2
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0answers
27 views

Bound on $E\left[f(Z,U)\exp \left(\frac{tZ^2}{2}\right) \right]$

Suppose $Z$ is standard normal and $U$ is another random variable. Let $f(Z,U) \ge 0$ also $ E[f^p(Z,U)] <\infty$ for $1 \le p <\infty$> we want to find a bound on the following expectation \...
2
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0answers
100 views

Functional invariance of exponential stochastic order

I am studying for an exam on stochastic order. I am struggling with a question on functional invariance of exponential order ($\leq_{\mathrm{e}}$), where for r.v.s $X$ and $Y$, $$X \leq_{\mathrm{e}} Y ...
2
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0answers
573 views

Moment generating function of $(W_T, \max W_t)$

Does there exist an explicit formula for the moment generating function $\psi(u, v) = E e^{u W_T + v M_T}$ of the pair $(W_T, M_T)$ where $M_T = \max_{0\leq t\leq T} W_t$? Using the well-known pdf of ...
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0answers
19 views

Is there a name for expressions that are invariant under the exchange of raw moments and cumulants?

I'm interested in expressions that are invariant under the exchange of raw moments and cumulants. This is trivially true of all expressions written only in terms of first order moments but nontrivial ...
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1k views

Moment generating function of a piecewise function

I am struggling like crazy with question b. Here is what I have come up with: But if I then test $M_X'(0)$, I don't get the expected value I calculated in part a: Where have I gone wrong?
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181 views

Moment Generating Function of a Beta random variable.

After getting some excellent help on this problem in the statistics SE, I am reformuluating my question. Let me know if I should just delete it and ask a new one. Let $V$ be a $Beta(\alpha,1)$ ...
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0answers
2k views

Moment-generating Function of a Continuous R.V. whose P.D.F is 1 from (0, 1)

I have been working on this problem for a few hours now and I feel I am missing something simple. The problem is to find the moment-generating function of a continuous random variable whose ...
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0answers
24 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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0answers
22 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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0answers
29 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 ...
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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}...
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0answers
76 views

Moment Generating Function - Cauchy Random Variable

I want to show that, for every $t\neq 0$, the moment generating function of the standard Cauchy distribution is equal to $+\infty$, i.e. $$M_X(t) = \int_{-\infty}^{+\infty} \frac{e^{tx}}{1+x^2}dx = +\...
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24 views

Transformation of RV's Moments

Given a transformation of the RV $X$, how do it's moments transform? More Detailed Formulation Suppose that $X:(\Omega,\mathcal{F},\mathbb{P})\rightarrow \mathbb{R}$, is a random variable whose MGF ...
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36 views

Establish bound for a probability using moment generating function

I have the following question Let $X_{1}$, $X_{2}$, ..., $X_{n}$ be independent and identically distributed random variables with moment generating function $M_{X}(t)$, for -h < t < h, ...
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81 views

Moment generation function - compound Poisson

Let $S_1$ be distributed with compound Poisson $\lambda_1=2$ and discrete indeminizations: $f_1 (x), x \geq 0$. And let $S_2$ be compound Poisson distributed with $\lambda_2=4$, and discrete ...
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0answers
20 views

Tight Bounds on Moments of $X = a_1x_1 + \cdots + a_nx_n$

Suppose that $x_i,z_i$ are i.i.d and that the moments $\mathbb{E}x_i^m \leq K \mathbb{E} z_i^m$ uniformly for $m=1,2,\ldots$. For simplicity's sake take $x_i$ and $z_i$ to be symmetric so we can ...
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0answers
40 views

Negative Binomal 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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54 views

Application of approximation of moments on poisson distribution

Exercises 101 in Chapter 4 Of the book "Mathematical Statistics and Data Analysis" by Rica states: Find the approximate mean and variance of Y = √ X, where X is a random variable following a ...