Moments are summaries of random variables' characteristics (e.g., location, scale).

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Proof of recurrence between cumulants and central-moments

According to Wikipedia, the $n$th cumulant $\kappa_n$ is related to the central-moments $\theta_n$ by the following recurrence: $$\kappa_n = \theta_n - \sum_{m=1}^{n-1} \binom{n-1}{m-1} \kappa_m \...
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Converting central moments to non-central moments (and back)

The central moments of a probability distribution $p(x)$ are defined as: $$\theta_n = \langle (x - \langle x \rangle)^n \rangle $$ while the non-central moments are the standard: $$\mu_n = \langle ...
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Absolute moment of Laplacian distribution

I need to compute the absolute moment of a Laplacian distribution. I just see in wikipedia that If X ~ Laplace(0, b) then |X| ~ Exponential(b ^ −1) where for exponential distribution $$E[X^n]=\...
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Moment Generating Function or Probability Distribution

I have a stochastic model and have calculated the higher order moments of the random variable through simulation. I want to find the probability distribution function that characterizes the model or ...
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Moments from pdf

I got confused reading about moments and their relationship with the pdf. Given a pdf and the values of the parameters, can we calculate the moments of the distribution? More importantly, what is the ...
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Why is the arithmetic mean smaller than the distribution mean in a log-normal distribution?

So, I have a random process generating log-normally distributed random variables $X$. Here is the corresponding probability density function: I wanted to estimate the distribution of a few moments ...
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Finding moments of the untruncated distribution from the moments of the truncated one

I would like to get the moments of a normal distribution from the moments of a two-sided truncated normal distribution in R. Their formulas are available on the truncated normal distribution ...
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Is the maximum entropy distribution the same for conditional and unconditional moments?

Suppose I have a set of observations drawn from some finite interval from a distribution that has a range that includes that interval but extends beyond it. This could be either because of the values ...
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How to calculate the central moment giving function of a distribution

Is there a function which gives the central moments instead of just moments of a distribution and if so how to calculate this function for a distribution e.g. the normal distribution.
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How would you calculate $E[\mid x \mid ^{\alpha }], \alpha \in \Re$?

Here $x \sim N(0,1)$. I realize that the expectation won't be defined for $\alpha$ when the integral goes to infinity. I can't seem to figure out which specific values of $\alpha$ would cause this. ...
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Creating a sample from multiple distributions and then finding deviation. What is the term?

I have 1000 gamma distributions, each with a different mean but constant ratio :$$\frac{mean}{s.d.}$$ also known as coefficient of variation. I extract one sample from each. Then I calculate : $$\...
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Unbiased estimator variance of sample variance

I was reading the section on k-statistics on wolfram alpha. It was known to me that for the sample variance $k_2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \overline{x})^2$ it holds that its variance ...
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Linear moment inequality for layman

Can someone please explain Manski approach to partial identification in very basic terms? I have gone through a lot of stuff but failed to find an intuitive explanation. I also don't understand how ...
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Does a scaling and shift of first two moments change higher moments too?

For a given random variable $X$ with mean $\mu_{\mathrm{old}}$ and standard deviation $\sigma_{\mathrm{old}}$ I would like to perform a transformation $g$ to obtain a new random variable $Y := g(X)$ ...
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If you know the central moments of the data $X$, find a function $f$ for which $f(X)$ has arbitrary central moments

Say you are given one-dimensional data $X$, with mean $\mu$ and central moments $a_n$ which you know. Can you construct a function $f(x)$ which transforms the data such that $f(X)$ has the central ...
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Why is a moment called a moment? [duplicate]

Someone told me that the term "moment" in Statistics comes from Physics. But I fail to understand how it relates to the definition of a moment of a force, which is a measure of its tendency to cause a ...
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Normalizing skewness with the Power or Box-Cox Transformation

Suppose I have a random sample drawn from an arbitrary strictly positive continuous distribution. Suppose moreover that I want to use the Box-Cox transform to zero out the skewness. Is there an ...
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How to combine statistical moments at different orders?

I am performing a number of simulations involving the estimation of statistical moments at different orders (lets say, from the 1st till the 3rd order). For certain case studies, I have useful ...
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What is $V(X^t)$ for any $t$ when only $E(X)$ and $Var(X)$ are known and $X$ is assumed normal?

Summary I'm trying to calculate $Var(X^t)$ where $t$ is the number of periods using only the following known parameters: $E(X)$ and $Var(X)$. $X$ is a random variable and is the return factor $(1 + ...
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Visualizing higher-order cross-moments (cokurtosis, coskewness)

How could and coskewness and cokurtosis be visualized in an easily comprehensible manner? Mean, variances, skewness, kurtosis can easily be illustrated in density plots: (Source: own *TeX-stuff) ...
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Moment of Z (about 0) in Arcsine distribution

Moments of Z (about 0) are $E(Z^n)=\int\limits_0^{\pi/2}\frac{2}{\pi}sin^{2n}\theta d\theta=\prod\limits_{j=0}^{n-1}\frac{2j+1}{2j+2}$ Would any stats expert explain me how the R.H.S is arrived at?
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Existence of the conditional tail mean

Does the existence of the first moment of a generalized Pareto distribution with support $[0,\infty)$ imply the existence (finiteness) of the conditional tail mean -- i.e. what in risk management is ...
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Regressions and expected value

Assume I have $Y=\beta_0 + \beta_1*X_1+u_0$ and $Y=\alpha_0+\alpha_1*X_1+\alpha_2*X_1^2+u_1$ where $E[u_0|X]=E[u_1|X]=0$ When is it true that $\alpha_1=\beta_1$? I did a sort of reversal proof: ...
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(2m-1)th Moments of GARCH(1,1) Process

I am currently studying the 2mth generating moment function of a regular GARCH(1,1) process which is given by Bollerslev (1986) as $$\mathbb{E}[\epsilon_t^{2m}] = a_m\bigg[\sum \limits_{n=0}^{m-1} a^...
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Binomial vs inverse binomial transform in obtaining moments

I am trying to obtain raw co-moment matrices from central co-moment matrices and I have found formulas for the transition from central to raw moments here . I have also found the opposite, that is ...
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Level-crossing and statistical moment

I recall that I read some time ago that the count of level crossings of a signal in a period of time is equivalent or proportional to a statistical moment for that signal (cannot recall the particular ...
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Binomial and Poisson issues (Jacod and Potter)

I've been reading through Probability Essentials by Jacod and Potter (2nd edition). I'm on a voyage to do every single exercise in the book. The following problems I am unsure of is as such: 5.11) ...
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How's product moment generating function different from moment generating function?

I have a reference that gives an example of a Poisson "product moment generating function": $$G(t)=E(t^X)=\sum_{i=0}^{\infty}t^i\frac{\theta^i}{n!}e^{-\theta}=\sum_{i=0}^{\infty}\frac{(\theta t)^i}{n!...
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What this slowly-varying nonstationary behavior of $x(n)$ implies for $y(n) = f[x(n)]$?

Folks, I am trying to figure something out here without success. Suppose $x(n)$ is a random discrete-time signal (or random time series) containing an arbitrary number of samples (say, $N$ samples). ...
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Kurtosis for circular data

This is a sister question to the one here. Again, I'm attempting to replicate a particular analysis of circular data. The author of this analysis writes I calculated . . . kurtosis for the ...
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Circular variance

I'm attempting to replicate a particular analysis of circular data. The author of this analysis writes I calculated . . . variance . . . for the observed recall errors, defining variance as the ...
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Determine distribution of a limit of random variables

Suppose a box contains a blue ball and green ball. After every hour a ball is chosen from the box randomly and then is put back with another ball of the same colour. After $h$ hours, there are $h ...
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Derivation of the third moment of the count joint statistic

Does anyone know where I can find the derivation of the third moment of the joint count statistic? I found this similar question answered in the past: need derivation of join-count variance (spatial ...
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Rule of thumb for number of observations required to estimate moments

I have a vague memory of reading about a rule of thumb for how many observations are required to estimate each moment. i.e. that to estimate the mean you need at least x observations, to estimate the ...
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Covariance for three variables

I am trying to understand how covariance matrix works. So let's suppose we have two variables: $X, Y$, where $\text{Cov}(X,Y) = \mathbb{E}[(x -\mathbb{E}[X])(y-\mathbb{E}[Y])]$ gives the relation ...
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How can I scale the $k$-th moment of a time series to a different time frequency?

I have a time series, let's say N daily log-returns. I want to study the moments (possibly the distribution) of the weekly returns. I have two ways: 1) Using the time-additivity property of ...
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Parametric distribution where the mean of a constant to the power of the random variable is an easy to use function

Let $X$ be a random variable with distribution $F\left(x;\theta\right)$ where $\theta$ are the parameters of the distribution. Let $c$ be a constant. Define: $h\left(c,\theta\right) = \mathbb{E}\...
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How to calculate $E(x)$ and $V(x)$ when $g(x)\sim f(g(x))$

Let assume that we are interested in a variable $x$. We know that e.g. $g(x)=x^2$, $g(x) \sim Uniform(a,b)$ or any other distribution. From that I can calculate $E(g(x)) = (a+b)/2$ and $V(g(x))=(b-...
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Are moments related to moments in physics? [duplicate]

The terms are the same, but I cannot outwardly see, whether moments in statistics have something to do with moments in physics. So are they related? How?
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What is the moment of a joint random variable?

Simple question, yet surprisingly difficult to find an answer online. I know that for a RV $X$, we define the kth moment as $$\int X^k \ d P = \int x^k f(x) \ dx$$ where the equality follows if $p = ...
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Higher-order cumulant and moment names beyond variance, skewness and kurtosis

In physics or mathematical mechanics, starting from position $x(t)$, one obtains rates of change via derivatives with respect to time: velocity, acceleration, jerk, jounce (4th order). Some have ...
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What does the MGF do when its parameter is other than 0?

I've only seen the MGF used when it takes 0 as it parameter and then derived for the nth moment. What does the MGF tell us when the parameter is other than 0?
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What is “t” in generating functions?

I am studying generating functions applied to probability (moment generating functions, probability generating functions and characteristic functions). I perfectly see their purposes and usefulnesses, ...
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Convergence of expectation

Suppose we have $X_n \overset{D}\to X$ for some sequence $X_{1},\dotsc, X_{n}$. Is it the case that if $E(X_{n}^2) \to E(X^2)$ we have it that $E(X_n) \to E(X)$, and when would it hold? My first ...
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What is the effect of taking the mean of a moments-based estimate?

Suppose I estimate the second and fourth moments of a signal as $M_2 \approx \frac{1}{N} \sum_{n=0}^{N-1} | y_n |^2$ and $M_4 \approx \frac{1}{N} \sum_{n=0}^{N-1} | y_n |^4$ and then I use these ...
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Approximate estimation of a covariance involving ratios

I have three random variables (T, O, A) that are each approximately normal. These three random variables are combined to form two ratios: X=T/A and Y=O/A. I wish to get an estimate of the covariance ...
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Empirical validation of a regression model estimating the mean and the variance?

I would like to empirically validate over a given dataset a regression model $\mathbb{R}^n \to \mathbb{R}^2$ that outputs both the mean and the variance. In particular, I am seeking a metric that ...
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First and second moments of deep nesting of the Binomial-Binomial hierarchical model?

I am interested in the Binomial-Binomial hierarchical model, where the number of trials itself follows a binomial distribution. I would like to know the expected value (first central moment, $\mu_1$) ...
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How to analyse a distribution with some given sample moments?

Suppose that I have n observations, $$X_1,...,X_n$$ with unknown distribution, n being small (say, between 6 and 20). If I know the first four sample moments (average, standard deviation, skewness, ...
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Expression for variation in the fourth moment of standard normal vs. sample size

Question: What is the expression for the variation in the fourth (not-centralized) moment as a function of sample size for the standard normal distribution and is there a proper analytic/symbolic way ...