# Questions tagged [moments]

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### Are the first 4 statistical moments independent? [closed]

Are the first 4 statistical moments independent? Is there a mathematical demonstration that can show independence one from each other? Can the value of one moment influence the value of another? If so,...
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### Is it always possible to determine the distribution of a random variable given all its moments? [closed]

we we're asked about it, and I know that answer is "NO", and I haven't found an good enough answer yet and would appreciate an explanation with examples.
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### The distance distribution of graphs

The degree distribution of a graph is of main importance, especially for large graphs, and namely random graphs. Its expected value and its higher moments tell a lot about a graph – but of course not ...
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### How to project a sequence on the univariate moment cone

Folowing closely Schmüdgen, K. (2017). The moment problem (Vol. 9). Berlin/New York: Springer., Chapter 10, Section 2, for a bounded interval $[a,b] \in\mathbb R$ and $m\in\mathbb N$, define the ...
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### Mixed inverse factorial moment of a multinomial random vector

Let $\mathbf{X}=(X_1,X_2,\ldots,X_k)$ be a multinomial random vector with parameters $n,p_1,p_2,\ldots,p_k$. I would be interested in computing mixed inverse factorial moments for $\mathbf{X}$. More ...
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### Is this (somewhat specific) moment problem treated somewhere?

Suppose I have a measure $\mu$ over $\mathbb R_+$ given by its moments $\mu_0,...,\mu_n$, defined as : $$\mu_k = \int x^{k} \partial\mu(x),\; k \in 1,...,n$$ Using Faà di Bruno's formula, I can ...
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### Moment-augment marginal distribution (not the exact name, given by myself)

Consider a joint distribution density function $f(x,y)$ of two random variables $x$ and $y$, and define $f_{n}(x)\equiv\int\!{\rm d}y\,y^n f(x,y)$. Do they have given name(s) in probability theory ? I ...
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### Moments of Logistic SDE's solution

On this article starting from equation $(30)$ it's presented a derivation of the first moment for the solution the logistic SDE: $$dx=x\left[\mu\left(1-\frac{x}{\tilde{x}}\right)dt+\sigma dW\right]$$...
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### Finding a r-atomic solution to the univariate truncated Hausdorff moment problem

Suppose i have a certain $t>0$, and observations of moments of a random variable $X$ given by $\mu_0=1,\mu_1,...,\mu_n$. How can i: Check that a measure with support $[0, \frac{1}{t}]$ with thoses ...
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### Cumulant of functions of weakly dependent random variables

Suppose $X_1,\dots,X_4$ are Gaussian random variables with mean and variance $$\mathbf E X_i = 0,\quad \mathbf E X_i^2=1.$$ Furthermore suppose that the random variables have a certain weak ...
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### What are the best unconditional bounds for the discrete moments of $\zeta'$?

I heard this morning through the app Researcher about the following article by Scott Kirila: arXiv:1804.08826v2 in which the author proves under RH that $J_{k}(T)\asymp_{k}(\log T)^{k(k+2)}$, where ...
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### Expected value of absolute value of shifted binomial distribution

Recently my research needs to calculate the close form of $\mathsf{E}[|X-\frac{n}{2}|]$ where $X$ follows binomial distribution with parameter $(n,p)$. When $p=\frac{1}{2}$, this is just the mean ...
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### A conjecture on 'truncated joint moments' of binomial coefficients under binomial distribution

This is similar in spirit to Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution but gives some total estimates. Though the other ...
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### $\mathbb{E}[X^4]=1$, $X,Y$ iid, what's the best upper bound of $\mathbb{E}[(X-Y)^4]$?

Let $X,Y$ be i.i.d. random variables, $\mathbb{E}[X^4]=1$, what's the best upper bound for $\mathbb{E}[(X-Y)^4]$ ? A trivial upper bound is $16$, since $(X-Y)^4 \leq 8 (X^4+Y^4)$ then take ...
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### Convergence rate of $\operatorname E|\langle X,f_n\rangle|^p$

Suppose that $X$ is a random element with values in a separable Hilbert space $\mathbb H$ such that $\operatorname EX=0$ and $\operatorname E\|X\|^2<\infty$. Suppose that $f_1,f_2,\ldots$ form an ...
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### Numerical evaluation/approximation of non-central high-order moments of high-dimensional Gaussian measures?

I need to numerically evaluate/approximate non-central high-order moments of high-dimensional Gaussian measures/distributions with given mathematical expectations and covariance matrices. The Gaussian ...