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Questions tagged [moments]

Moments are summaries of random variables' characteristics (e.g., location, scale). Use also for fractional moments.

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Proving that mgf determines distribution via Laplace transform

I am reading this question and the answer provided there about the moment generating function (mgf) and how its uniqueness can be proved via the uniqueness of Laplace transforms. In my book, Measure ...
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Second moment of weighted average of random variables

I stumbled upon problem 254 from the SOA Exam P list in https://www.soa.org/globalassets/assets/Files/Edu/edu-exam-p-sample-quest.pdf for which I am puzzled by the solution described in https://www....
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Relation between first three moments of distribution

Consider a random variable $X$ with distribution described by the first three moments: $$\mathbb{E}(X) = \mu$$ $$\mathbb{E}(X-\mu)^2 = \sigma^2$$ $$\mathbb{E}(X-\mu)^3 = \gamma$$ Is there a nontrivial ...
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Estimating correlation parameter from known value of bivariate normal distribution

I want to estimate the correlation parameter $\rho$ using the following expression taken from this paper (equation 10 on page 17): $$ \hat{s}^2+\hat{\mu}^2=N_2(N^{-1}(\hat{\mu}),N^{-1}(\hat{\mu}), \...
MysteriousBrit's user avatar
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Higher order moments to evaluate strength of linear relationship between variables

Let $X_1,\dots,X_n$ be real random variables such that $\alpha_1X_1+\dots+\alpha_nX_n=0$ for some unknown $\alpha_1,\dots,\alpha_n$. If $n=2$, one can study the strength of linear relationship by ...
12345's user avatar
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By means of what distribution can I match the first n moments for arbitrary (i e. any) values of those moments?

Suppose I have the first n moments from some data set, either raw, centered or scaled, (or cumulants instead) whichever is more convenient for matching. Is there a continuous, continuously ...
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How can I implement moment matching with kernel tricks if I do not have the complete distribution but only the higher-order moments or culuments?

As is said in Appendix B.3 of ref, "It is difficult to match high-order moments, because we have to deal with high order tensors directly. On the other hand, MMD can easily match high-order ...
Kaiming Zhang's user avatar
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Moment Ratios and L Moment Ratios:

I want to know the difference between moment ratio diagrams and L moment ratio diagrams. As I understand the moment ratio diagram serve the purpose of detecting the type of probability density ...
profdr's user avatar
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6 votes
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166 views

Terminology clarification about sample moments

According to MathWorld (link): "The sample raw moments are unbiased estimators of the population raw moments". While in Wikipedia (link) it is said: ...the $k$-th raw moment of a population ...
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$\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$, upper bound second moment from first moment

Let $X$ be a non-negative random variable bounded on $[0,1]$. Is it true that $\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$ for some constant $k$? If not, are there any minimal assumptions on $X$ where this ...
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Why do the skewness and kurtosis formulae have powers of the variance in the denominator?

We calculate the variance as the centered 2nd moment $E[(X-\mu)^2]$. So when it comes to the skewness and kurtosis, why are the 3rd and 4th moments divided by the 3rd and 4th powers of $\sigma$? Why ...
ahron's user avatar
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what is the formula for calculation of fourth raw moment (or central moment) from variance and 3rd central moment (or raw moment)?

Basically the title. I can't seem to find any solution for this. I have the mean, variance or the second central moment and third central moment and third raw moment. I need to find the fourth raw ...
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Characterize conditions in which Taylor moment approximation is good

I am working with the Projected Gaussian, or Angular Gaussian distribution, which is given by $z = \frac{x}{||x||}$, where $x \sim \mathcal{N}(\mu, \Sigma)$. This is a distribution on the sphere in $\...
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7 votes
1 answer
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Recurrence formula for the moments of a half-gaussian distribution (on R+)

I am trying to compute an integral that looks like the moments of a Gaussian $\mathcal{N}(\mu, \sigma^2)$, but the main difference is that we only integrate over R+ and not R. I believe we could call ...
Julia Linhart's user avatar
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properties of a expectation for a non-negative random variable

Say I have a non-negative discrete random variable $X$ (values of $X$ can be mapped to integers $(0, 2^n -1)$ for $n \in \mathbb{Z}$) and an associated distribution $P(X)$. Given a non-negative scalar ...
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Relating Moments to OLS

I am trying to see the relationship between OLS and Method of Moments. Moment Equation: For a discrete random variable and a continuous random variable centered around some point "c": $$E[(X-...
Uk rain troll's user avatar
2 votes
1 answer
110 views

Simulate a distribution from a fitted beta-regression model for a density plot in R [duplicate]

I have produced the following figure by simulating some values from a fitted gamma regression with a low AIC value that provides the closest approximation of my raw data out of all of my models, and ...
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2 votes
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Distribution supported on $(0,\infty)$ for which moments of its truncated distribution are elementary functions of the truncation point and power

I am looking for a distribution with a differentiable PDF $f:(0,\infty)\rightarrow (0,\infty)$ for which for any $\delta>1,z>0$, the two following integrals are finite elementary functions of $\...
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Numerical moments of a multivariate Poisson Log-normal posterior

I have a log-density of the form: $$P(\mathbf{x}) \propto \exp\left( - \mathbf{b}^{\top} e^{ \mathbf{x} } - \frac{1}{2}\mathbf{x}^{\top}A\mathbf{x} \right)$$ where $A$ is a symmetric positive definite ...
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1 vote
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Taylor approximation for function of a random variable [closed]

There is a function $f$ whose domain is the space of CDFs on $\mathbb{R}_+$ and whose range is $[0,1]$, e.g. $f$ maps a CDF on to a real number. Further, $f$ is continuous, increasing with respect to ...
user's user avatar
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How can we efficiently find the fourth moment of a Poisson distribution?

Suppose we have $X\sim \textrm{Poisson}(\lambda)$ and we know that moment generating function $M(t)=\mathbb{E}(e^{tX})$. How do we use the moment generating function property $M^k(0)=\mathbb{E}(X^k)$ ...
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System GMM Estimator

Consider the dynamic linear model given by: \begin{equation} y_{it} = \rho y_{i,t-1} + \alpha_i + \nu_{it} \end{equation} where $\alpha_i$ represents individual fixed effects. The GMM two-step ...
Rebecca 's user avatar
5 votes
1 answer
199 views

expectation value, distribution function and the central limit theorem

The problem goes thus: ${\{X_n\}}$ is an $iid$ sequence of random variables with mean 0 and variance $\sigma^2$. If the third moment is finite, show that $$\lim_{n \to \infty} \mathbb{E} \left(\left(...
Snowflake's user avatar
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How do continuous partial derivatives depend on $n$ in maximum likelihood estimation?

I'm reading Tensor Methods in Statistics by McCullagh 1987, (P209 for this question) and I can't understand one step he uses. He begins with the usual log-likelihood \begin{equation*} l(\theta; Y) =...
Nick Green's user avatar
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Taking expectation of higher moments using original distribution [duplicate]

If we want to calculate $\mathbb{E}[X]$, we know this is $\int_{-\infty}^{\infty} x f_X(x) dx$. When we want to calculate $\mathbb{E}[X^n]$ (for, say, positive integer $n$), we also do this with ...
David's user avatar
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1 answer
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Higher order central moments for uncorrelated random variables

Two random variables (RVs) $x$ and $y$ are uncorrelated, which implies that $\mathbb{E}[xy]=\mathbb{E}[x]\mathbb{E}[y]$. Let the zero-mean deviation variable be marked with a tilde, e.g. $\tilde{x}=x-\...
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2 votes
1 answer
116 views

Moments of sum of squares of independent gaussians $X_i \sim \mathcal{N}(\mu_i,\sigma^2_i)$, or $||X||^2$

Say that we have $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$. Is there some formula to calculate analytically the expected value of the sum $S = \sum_i^n X_i^2$?. This is equivalent to computing $\...
dherrera's user avatar
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2 votes
1 answer
48 views

Pinsker-type inequality for moments?

Let $f_1$, $f_2$ be two discrete probability distributions. By Pinsker's inequality, the Kullback-Leibler divergence $D(f_1||f_2)$ sets an upper bound on the total variation distance between the two ...
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11 votes
2 answers
391 views

Formulas, approximations, or bounds for $\mathbb{E}\left( \frac{X}{\lVert X \rVert} \right)$, $X\sim N(\mu, \Sigma)$?

In another question, I asked for $\mathbb{E}\left( \frac{X}{\lVert X \rVert} \right)$, in the case where $X \in \mathbb{R}^d \sim N(\mu, I_{d})$. Somebody posted an exact formula based on the symmetry ...
dherrera's user avatar
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2 votes
0 answers
120 views

Variance of Fourth Sample Central Moment [closed]

I am trying to derive a formula for the variance of the fourth sample central moment $m_4=\frac{1}{n}\sum_{i=1}^n (X_i-\bar{X})^4$ (where $X_i$ is the $i$th realization of a random variable, $\bar{X}$ ...
Hiro's user avatar
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4 votes
1 answer
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Formulas or approximations for $\mathbb{E}\left( \frac{X}{\|X\|} \right)$, $X\sim N(\mu, Id)$?

This is a cross-posting of this math SE question. I want to compute or approximate the following expected value with some analytic expression: $\mathbb{E}\left( \frac{X}{||X||} \right)$ , where $X \in ...
dherrera's user avatar
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Estimating the population parameters using the sample moments

I have distribution , which is not normal and highly skewed. I take a sample from that distribution and calculate the first four moments of the sample distribution. The first four moments calculated ...
Machine_leaning_9's user avatar
4 votes
1 answer
143 views

Minimum Pearson's correlation between $X$ and sign($X$)$\cdot X^2$

Suppose $X$ is a (whatever bounded or not) continuous random variable ($X$ is not constant) with an arbitrary distribution. Is it possible to construct a distribution s.t. $\text{Corr}(X, \text{sign}(...
cat's user avatar
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4 votes
1 answer
200 views

Unbiased estimators and moment of moments

Following section 7.4 of Rose and Smith "Mathematical Statistics with Mathematica" (book available online here), I'm trying to use the Fundamental Expectation Result (eq 7.15) and other ...
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3 votes
1 answer
90 views

What is the fourth moment of a Euclidean Norm?

Let $X=\lVert M^\top p\rVert_2$, where $M$ is an $n\times n$ non-random matrix and $p\sim N(0,I_{n\times n})$ is an $n\times 1$ vector, and$\lVert \cdot\rVert_2$ is the Euclidean norm. Using some ...
Carl's user avatar
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0 votes
1 answer
155 views

The third central moment of a sum of two independent random variables

Is it true that in probability theory the third central moment of a sum of two independent random variables is equal to the sum of the third central moments of the two separate variables?
AdVen's user avatar
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Given two rvs $X$ and $Y$, if $X Y = Z$, is it possible to change the mean and sd of $X$ without changing the mean and sd of $Y$ and $Z$

I have two lognormal rvs $X$ and $Y$, and a third rv $Z$ which is the product of the former two. I know the mean and standard deviation of the three. Is it possible to calculate an alternative pair of ...
Pau's user avatar
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1 vote
0 answers
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Covariance of square root of random variables [closed]

Suppose I have the following expressions, where $X_1$,$X_2$,$Y$,$Z_1$,$Z_2$ are all random variables. $$X_1^2 = B_1 Y + B_{Z_1} Z_1$$ $$X_2^2 = B_2 Y + B_{Z_2} Z_2$$ I'm interested in $Cov(X_1,X_2)$. ...
kerfelafel's user avatar
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54 views

Variance of powers of a standard normal random variable

To predict growth of money in a stock market I try to calculate expected return over a longer timeframe (e.g. 30 years) with a confidence interval. The simple math of taking an average stock market ...
Bastiaan's user avatar
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0 answers
122 views

Understanding relation between axis of least and maximum second moment

I was going through computer vision lecture video. You can find the pdf of this lecture here. I was trying to understand how orientation of object corresponds to axis of least second moment aka ...
Mahesha999's user avatar
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30 views

Expectations with respect to affine transformation of a log-normal distribution

Let $X$ be a log-normal distribution and consider $Y=aX+b$ for some $a,b>0$. I would like to know if one can compute $$\mathbb{E}[\log(Y)]$$ This would be very easy if it was $b=0$, since in this ...
Francesco Bilotta's user avatar
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0 answers
27 views

Characterizing/Estimating heaviness of tails using ratio of moments

For any probability distribution function (PDF), $p(x)$, which has finite moments $\left<X^k\right>$ defined upto $k=N$, is it possible to say something about the heavyness of the tails by ...
user35952's user avatar
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6 votes
2 answers
171 views

Can positive values with sd > mean have skewness = 0?

I'm trying to create an example of a distribution with all positive values, standard deviation > mean, and skewness =0 (third moment). I cannot. Is that possible? Can you prove it mathematically? ...
GabyLP's user avatar
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1 vote
0 answers
41 views

Can the logit normal distribution be uniquely defined by its first two moments? [closed]

I was wondering if the logit-normal distribution is uniquely defined by its first two moments; I believe this to be the case as the logit-normal distribution is simply the logit transform of a normal ...
user391050's user avatar
3 votes
1 answer
73 views

Calculating Kurtosis for Groups Containing Fewer Than 4 Observations

Based on some preliminary exploration, here are some interesting observations about kurtosis for when you're calculating kurtosis for groups that have fewer than 4 observations. First, here's the ...
David Moore's user avatar
3 votes
1 answer
171 views

Does the Central Limit Theorem Apply to All Finite Samples Even If They Come From Distributions That Don't Have a Finite Variance?

Some distributions, like the Cauchy distribution, don't have a finite variance, and therefore the central limit theorem does not apply to them. If I have a thousand randomly selected observations from ...
David Moore's user avatar
3 votes
1 answer
77 views

Minimum Numbers of Observations for Standardized Moment Calculations

You can take the mean of any number of values, including just one value - in that case, the mean will just be equal to that value. Standardized means (standardized first moments) are always equal to ...
David Moore's user avatar
1 vote
0 answers
55 views

Does convergence of moment implies existence of finite moment? [closed]

Let $X_n$ by a random variable that converges in distribution to $X$. On the top of that, $$\lim _{n \rightarrow \infty} E[X_n] = E[X] = O(1).$$ Does it implies that for every $n$ $$ E[X_n] < \...
Eryna's user avatar
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1 vote
0 answers
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Bayesian VAR: Derivation of predictive distribution for reduced form VAR

I have a standard reduced form VAR of type without intercept: $y_t = A_{1}y_{t−1} + \ldots + A_py_{t−p} + e_t$, $e_t \sim N(0,Σ)$. I need to derive the predictive posterior distribution $p(y_{T+h}|y_{...
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0 answers
52 views

Central limit theorem : relaxing assumption of all finite moments

Consider $S_n = \sum_{i = 1}^n b_{i,n} X_{i,n}$ where $X_{i,n}$ are random variable neither independent neither identically distribution and $b_{i,n}$ are weights satisfying the Lindeberg condition. I ...
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