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Questions tagged [moment-problem]

The moment problem arises as the result of trying to invert the mapping that takes a measure $μ$ to the sequences of moments, and to resolve the problem of determinacy of such measure.

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86 views

If all powers of two random variables are uncorrelated, are they independent?

Let $X$ and $Y$ be random variables on a common probability space. If $$\def\E{\mathbb E}\E[X^nY^m]=\E[X^n]\,\E[Y^m]<\infty $$ for all integers $n,m\ge 0$, does it follow that $X$ and $Y$ ...
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11 views

Moment in parallelogram

If F is a force in the same plane as parallelogram ABCD and the moment of F about A equals -18 moment unit and the moment about B equals the moment about D equals 32, what is the moment of F about C? ...
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24 views

Do orthogonal polynomials determine the moments of their orthogonality measure?

I am currently learning about the inverse problem for orthogonal polynomials for orthogonality measures supported on the real line. My question is not about finding the orthogonality measure from the ...
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113 views

Constructing a probability measure on the Hypercube with given moments

Let $H = [-1, 1]^d$ be the $d$-dimensional hypercube, and let $\mu \in \text{int} H$. Under these conditions, I can explicitly construct a tractable probability measure $P$, supported on on $H$, ...
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48 views

Sample instances of random process given all temporal correlation functions?

I asked this question on signal processing stack exchange (question) but I wonder the general answer I am seeking makes the question better suited here. Suppose I have a complex valued random process ...
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30 views

Discrete positive moment problem

My teacher claims that, given all factorial moments $E((X)_r) = E(\prod_{i=0}^{r-1}(X-i))$ of a positive discrete random variable $X$ it is possible to deduce the law of said variable. The first ...
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1answer
68 views

Explanation of Proof of Theorem 6.1 of Chapter II of The Laplace Transform by Widder

I was reading The Laplace Transform by Widder and have a problem in understanding the proof of theorem 6.1 in chapter II. Please see the following image I am unable to understand how equation 2 ...
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1answer
51 views

What conditions on the moments make a measure a probability measure?

For a positive Borel measure $\mu$ on the real line, let $\displaystyle{m_n = \int_{-\infty}^\infty x^n d\mu(x)}$, i.e. the $n$th moments of the measure. Are there any conditions on $m_n$ for when $\...
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24 views

$\int x^k\ d\mu(x)\geq0$ for $k$ even?

I am studying convex optimization, and the truncated moment problem is being discussed. I have no background in measure theory so I don't understand things taken for granted such as $$\int x^k\ d\mu(x)...
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2answers
15 views

Expected value of half normal using gamma function

Let $X$ be $N(0,1)$. I want to compute $E(|X|)$. This is what I tried: $$E(|X|)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty|x|e^{\frac{-x^2}{2}}dx=\frac{2}{\sqrt{2\pi}}\int_{0}^\infty xe^{\frac{-x^2}{...
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1answer
23 views

Find the Distribution that corresponds to the given MGF

I am working on a problem and am a little bit confused. I need to find the distribution that corresponds to the MGF: $2e^t\over3-e^t$ Do we need to separate this into something like: 2e$^t$ and $1\...
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1answer
17 views

Finding the Law of X

I am woking on a problem and am unsure how to approach it. It is finding the law of X (the distribution that correspond to) for MGF $2e^t\over3-e^t$ I am thinking that this is perhaps an exponential ...
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4answers
43 views

Finding MGF of Multiple RV

I am working on a problem and am a bit stuck. The problem: Let X1, X2, X3 be i.i.d random variables with distribution P(X1=0) = $1\over3$ P(X1=1) = $2\over3$ Calculate the MGF of Y = X1X2X3 Not ...
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2answers
59 views

Finding Standard Deviation from MGF

I a working on a problem and am a little bit confused at how to approach solving it. The problem: Given the MGF F(t) = $1\over(1-2500t)^4$ Calculate the SD. Do we need to do some substitution with ...
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30 views

About the $k$th moment of the sum of independent r.v.

It is known that $$ E[\:\{\:(X+Y) - E(X+Y)\:\}^2\:] = E[\:\{\:X - E(X)\:\}^2\:] + E[\:\{Y - E(Y)\:\}^2\:] $$ (that it is the variance of the sum is the sum of variances given that the r.v's are ...
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122 views

Calculating a bias given the MOM and MLE

Question: A company manufactured objects, starting from 1 to N. One of the objects is selected, and serial number is 888. We are asked to find things like the MOM, MLE etc. which I managed without ...
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How are moments used in representing differential equations?

I keep seeing the use of 'moment closure' techniques in papers, but I've never found any material explaining how/why 'moment closure' is being applied. I get that there's a differential equation that ...
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1answer
147 views

Kurtosis poisson

I am trying to find the kurtosis of the Poisson - I found E(X 4)) = λ4 + 6λ3 + 7λ2 + λ` The 4th central moment I calculated with the binomial expansion: E(X 4)) - 4 E(X 3)) E(X) + 6E(X 2)) E(X)2 ...
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1answer
52 views

Maximizing expected value with constrained 2nd moment

$$\begin{array}{ll} \text{maximize} & \displaystyle\int_{0}^{1} x \, f(x) \, \mathrm dx\\ \text{subject to} & \displaystyle\int_{0}^{1} f(x) \, \mathrm dx = 1\\ & \displaystyle\int_{0}^{1} ...
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2answers
14 views

Density tranformation theoren, n=1 - exercise solution

Let X be uniform on $[0,1]$, and let $Y=\sqrt{x}$. Find $E(Y)$ by a) finding density of $Y$ and then finding the expectation, and b) by using definition $\mathbb{E}[g(x)]=\int g(x)f(x)dx$. ...
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1answer
85 views

moments estimation using Rayleigh distribution

consider Rayleigh distribution: $f(x;\theta) = \frac1\theta e^{\frac{-x^2}{(2\theta)}}$, x > $0$ and $\theta$ > $0$ Show that $E(X^2) = 2\theta$ On the basis of the proceeding item, construct an ...
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49 views

Delta method application

I have been trying to understand the delta method, with no success. Our professor gave us an exercise (with solution), however I do not even know where to start solving the problem. Question: ...
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25 views

What can be said about the set of distributions with given moment sequence?

It is a classical result that a sequence of moments not necessarily uniquely characterises the distribution. I would like to know if there are any results telling me something about all distributions ...
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1answer
139 views

Moments of products of independent random variables: $E[ X^kY^k ]$

If we are given that two random variables $X$ and $Y$ are independent, I'm wondering if the rule: $E[XY] = E[X]E[Y]$ applies for any integer $k>0$, such that: $E[X^kY^k] = E[X^k]E[Y^k]$. Is ...
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Higher moments are minimized around WHAT point?

It is well known that the 2nd moment (physics: moment of inertia) is minimized when centered around the mean (physics: center of mass). However, I recently realized that higher moments are NOT ...
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1answer
93 views

Which moments identify an absolutely continuous measure on the unit circle?

Suppose you have a sequence $(s_k)_{k\in\mathbb N}$ of complex values such that there exists a finite measure $\mu$ on the complex unit circle with $s_k$ as its moments $$ s_k = \int_{\mathbb T} z^kd\...
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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
95 views

Does the condition $E[X]=E[X^2]=E[X^3]$ determine the distribution of $X$?

This is a question out of pure curiosity, motivated by this posting. Here I checked that if a $\mathbb{R}$-valued random variable $X$ has finite $4$-th moment and $E[X^2]=E[X^3]=E[X^4]$ then $X$ is a ...
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87 views

Moment of Inertia of a rotated quarter circle

I'm trying to find the moment of inertia of just the quarter circle part for the x1, x2 axis, which I believe would be (being $\rho$ the density): $$\frac{\rho}{16}\pi+\frac{\pi\rho}{4}(1-\frac{4}{3\...
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22 views

Discrete, finite L-moment problem

Suppose that we have a real-valued discrete random variable, whose probability distribution has finite support on some set $S$ of real numbers. Then if $N = |S|$, we know that we can construct the ...
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1answer
34 views

Sufficient condtion for the existence of E[|XY|]

I have read a claim that $E[X^4]<\infty$ and $E[Y^4]<\infty$ imply that $E[(XY)^2]<\infty$ but I cannot prove it. $X$ and $Y$ are not independent and they are correlated.
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1answer
686 views

Method of moments estimator question given PDF

I'm having a little trouble with this question which I got from my statistics tutorial. If $Y_1, Y_2, ..., Y_n$ is a random sample from a population Y having a probability density function given by: ...
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245 views

Variance for the function $g(x)=\frac{1}{(1.1-x)^2}$ where x is has uniform distribution $U(0,1)$

Calculating the mean of the random variable $g(x)=\frac{1}{(1.1-x)^2}$ where x is has uniform distribution $U(0,1)$: $E[g(x)]=\int\limits_{0}^{1}\frac{1}{(1.1-x)^2}dx=\frac{1}{0.1}-\frac{1}{1.1}=9.09$...
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2answers
119 views

found mgf, but $E(x^3)$ is not defined

A question asks me to find the MGF for the continuous random variable $f(x) = 3x^2$ on $[0,1]$. Using the MGF, it asks me to find $E[X^3]$. I'm having trouble evaluating $E[X^3]$ I found the MGF ...
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2k views

Moment of force acting at a point about other point

$F=4i-2j$ acts at a point $A=(2,5)$ then the perpendicular distance between $B=(10,3)$ and line of action if the force $F=..$? I found this kind of questions disturbing I don't know if I should start ...
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65 views

Determining a PDF from its moments

I have a probability density function, $f_{X,Y}(x,y)$. I can calculate moments of $f$ easily, namely, I can calculate $\int\int x^ay^bf_{X,Y}(x,y)\,dx\,dy$ over the range of $X$ and $Y$, for any ...
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1answer
52 views

Why study moment problem in one dimensional case?

I have been reading about moment problem and I have been curious about the following question. What is the motivation for studying the Hamburger moment problem(one dimensional moment problem? I ...
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1answer
219 views

Orthogonal polynomials of the second kind

Let $L: \mathbb{R}[x] \rightarrow \mathbb{R}$ be a positive definite linear functional and let that $\{s_n\}$ be a positive semi-definite sequence such that $$L(x^n)= s_n, n\ge 0$$ and $$<p,q> =...
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1answer
30 views

Uniqueness of a moment problem over $\int^{\infty}_{0} \frac{\sigma^p}{(1+\sigma)^{2S}} m(\sigma) \: \mathrm d \sigma$ with a finite range in $p$

A question over on the physics site asked about the moment problem $$ \int^{\infty}_{0} \frac{\sigma^p}{(1+\sigma)^{2S}} m(\sigma) \: \mathrm d \sigma = \frac{p!(2S-p)!}{(2S+1)!}, $$ where $S=0,1,2,\...
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130 views

Existence of random variable given first $k$ moments

A sequence of real numbers $\{m_k\}$ is the list of moments of some real random variable if and only if the infinite Hankel matrix $$\left(\begin{matrix} m_0 & m_1 & m_2 & \cdots \\ ...
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99 views

Stochastic Process Moment Problems

Given a stochastic process $X:S\times \Omega\rightarrow \Bbb{R}$ with $S\subset \Bbb{R}^d$, one can define its moment functions as follows (when they exist): $$ M_{\boldsymbol{\alpha}}(t_1,\ldots,t_n)...
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46 views

Moment of measure for Dirac disintegration

Let $X,U,V$ be three subsets of $\mathbb{R}^n,\mathbb{R}^a,\mathbb{R}^b$ respectively and $\mu$ a probability measure supported on $X\times U\times V$. The marginal $\mu_X$ of $\mu$ w.r.t $x$ is the ...
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41 views

Method of moments estimate: custom distribution

Find the maximum likelihood estimation of $\mu$ and the population mean given a $n$-sized sample of a population with distribution function: $$\begin{aligned}f(x)=\frac1p, \quad 0 < x < p \...
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49 views

Calculus 2 moments question.

Calculate moments $M_x$ and $M_{yc}$ of mass of the lamina with density $p=2$ that is the shape of a quarter circle, centered at the origin, in the first quadrant. Note: $yc$ denotes $y$-centre. I ...
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1answer
474 views

Positive definite sequence and its corresponding determinant.

I am currently reading the book of Akhiezer http://www.maths.ed.ac.uk/~aar/papers/akhiezer.pdf and I saw on page 1 that a sequence $\{s_n\}_{n=0}^\infty$ of real numbers is positive definite if $$ \...
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30 views

Sequence to measure

Let consider a sequence of real numbers $(y_k)_{k\in \mathbb{N}^n}$. Take an order $r\in \mathbb{N}^n$ such that $k\le r$. Suppose that this numbers come from moment of a measure $\nu$ supported on ...
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1answer
136 views

What is the variance of an estimator in terms of central moments

I've been asked to define an estimator, $\Theta_c$ for $\sigma^2$, where $c>0$ is a constant. $\Theta_c = c \sum_{i=1}^n(X_i-\bar{X})^2$ The question is to calculate Var($\Theta_c$) in terms of ...
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1answer
49 views

Question about Humburger moment problem and Characteristic function.

Let $\phi(t)$ be some a contionous infinitly differentialbe function such that $\phi(0)=1$ and $\phi(t)$ is symmmetric. Let \begin{align} m_{2n} =i^{-2n} \phi^{n}(0) \end{align} Suppose, that \...
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2answers
187 views

Can one tell based on the moments of the random variable if it is continuos or not

Suppose we are given moments of a random variable $X$. Can we determine based on this if the random variable is continuous or not? We also assume that the moments of $X$ completely determine the ...
2
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1answer
55 views

If $\lim_{ k \to \infty}\left (E\left[\vert X \vert^k\right]\right)^{\frac{1}{k}} =\infty$ then $X$ is unbounded [closed]

Give a random variable $X$ if we have that \begin{align} \\\lim_{ k \to \infty}\left (E\left[\vert X \vert^k\right]\right)^{\frac{1}{k}} =\infty \end{align} Does this mean that $X$ has unbounded ...