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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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Higher order moments of a multivariate Gaussian rv

Let $X~N_d(\mu,\Sigma)$ be a multivariate Gaussian random vector. Is there a convenient formula for each of $$ \mu_p\triangleq \mathbb{E}\left[\sum_{i=1}^d |X_i|^p\right], $$ in terms of $\mu$ and ...
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Measure modality of a distribution

In statistics, different orders of moments are tools to characterize a distribution, for example mean, covariance, skewness etc., which also gives an intuitive way to visualize the distribution. But ...
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Getting skew normal parameters from its moments

I have data on expenditure in dollars, and for set of countries i know average expenditure, sd, skewness. For example in country A mean=200\$, sd=100, skewness=1.5 and i want to estimate probability ...
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37 views

Determine expected value for continuous random variable [duplicate]

I know that $E[X^n]$ is found by $$\displaystyle\int_{0}^\infty{x^nf_x(x)dx}$$ I simplified this to $$\displaystyle\int_{0}^\infty{ \frac{x^{\frac{v}{2}-1+n}e^{\frac{-x}{2}}}{\displaystyle\int_{0}^\...
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Higher moments of linear regression residuals?

I previously asked this on Math StackExchange, with no success, but this post will add to that with some simulations. Background In the following linear regression with i.i.d $\epsilon_i$ $(i = 1, \...
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1answer
29 views

Relationship between exponential families and moment generating functions

I have recently been playing around with some change of measure arguments for shifting the mean of a sub-gaussian distribution. It occurred to me however, that sub-gaussianity might not be the natural ...
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53 views

Are there always two degrees of freedom in any probability distribution?

Take any random variable $X$ that follows some distribution $P(X)$. I was looking at this Wikipedia page and I'm trying to get some intuition for why we choose to define standard moments the way we do....
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Getting parameters of new distribution after moment matching using expectation propagation

In this paper, the authors apply Expectation Propagation for feature selection. However, I don't understand how to get the analytical expression from equation 25. They say: The right-hand side of (...
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What's the name of the ratio between successive moment of random variable?

Let $X$ be a random variable. Is there a name to the ratio: $\frac{E[X^{n+1}]}{E[X^{n}]}$ where $E[\cdot]$ is the expectation of $\cdot$. We let $n\ge0$ and for $n=0$ it's simply the expectation ...
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Gaussian distribution: moments, independence and rotation

I have a few questions with respect to the gaussian distribution, its moments and independence. So a gaussian distribution is fully specified by its first two moments, the mean and variance (or ...
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30 views

Minimum possible correlation for monotonic data

Suppose we have a dataset $(x_i, y_i)$ where both $x$ and $y$ are monotonically non-decreasing with $i$. So obviously the Spearman rank correlation between $x$ and $y$ is 1. However, what is the ...
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Estimate Information Entropy from Moments

I hope I'm using the right terminology below. I have access to the moments statistics of a large sample. That is, I have $\sum(x)$, $\sum(x^2)$, ..., $\sum(x^k)$. I also have access to max and min, ...
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109 views

Relations between moments and cumulants

From the definition of KGF (cumulant generating function) we can write: \begin{align} K_x(t) &= \log_eM_x(t) \\ &= \log_e\left[1+\sum_{r=1}^{\infty}\frac{t^r}{r!}\mu_r^{'}\right] \\ &= ...
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Moments of the Generalized Dirichlet distribution

I have been trying to solve the following integral. $ \int\theta_j \sum_{k=1}^K \theta_k \beta_{k,w} \prod_{k=1}^K \frac{\Gamma(\alpha_k + \beta_k)}{\Gamma(\alpha_k)\Gamma(\beta_k)} \theta_k^{\...
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Why Does Second Order Weak Stationarity include Statement on Covariances in addition to Statement on Mean and Variance?

A stochastic process is second order weakly stationary if all random variables have same mean (first moment), and same variance (second moment?), and covariances that are time-invariant (second moment ...
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Almost Gaussian distribution with interesting moments

I have a rather funny situation here. There is this random complex number, $z$, with an unknown distribution $\rho_\lambda(z)$ which depends on a parameter, $\lambda$. I can compute the moments and ...
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Can I use moments of a distribution to sample the distribution?

I notice in statistics/machine learning methods, a distribution is often approximated by a Gaussian, and then that Gaussian is used for sampling. They start by computing the first two moments of the ...
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In general, how should we find the pmf given only the moment generating function without comparing its form to that of famous pmf?

Background It is known that moment generating function generates moments, but does it hold information about the probability of the random variable being realised at a particular value? Example ...
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How to optimize ratiometric loss function with variance term in it?

I'm training a neural network (or any ML model with non-convex gradient-based optimization) to predict a continuous outcome variable. Currently, I use the mean squared error loss function, i.e., if $y$...
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1answer
52 views

Tradeoffs of robust mean measures (trimmed, Huber, cosh, etc)

After recently having delved into the world of robust measures (for location, mean being the classical case), I have had difficulty understanding robust measures' core dynamic. Basically, what are ...
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2answers
109 views

Testing whether the conditional correlations/covariances differ between two groups

I have two samples of variables $\{y_{1i},y_{2i},x_i,s_i\}$. Where $y_1$ and $y_2$ are binary variables, $x$ is a continuous variable and $s$ is a sample indicator, taking the value 0 in one sample ...
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Are there forms of error which affect kurtosis?

I am currently taking a research methods course online. Today we talked about systematic and random error. The instructor pointed out that systematic error is expected to influence the mean of an ...
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Parameter Estimation with Method of Moments

I was working on some problems and came across this one I was having trouble with: Given a random sample {0.1, 0.4, 0.2, 0.1} from a Beta population with β = 1, use the method of moments to find an ...
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$\mathbb{E}[\min (X_{1:n}) + \max(X_{1:n})]/2 = \mathbb{E}[\text{median}(X_{1:n})]$?

Say $X$ is continuous random variable, and we have $n$ iid samples, denoted as $X_{1:n}$. Then can we say the following $$\mathbb{E}[\min (X_{1:n}) + \max(X_{1:n})]/2 = \mathbb{E}[\mathrm{median}(X_{...
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Which concentration inequalities apply when moments are infinite?

I have 2 questions: Suppose I have a finite mean but an infinite variance for a discrete distribution w/support $\{1,2,\dots\}$. Is there any probability inequality tighter than Markov in this case? ...
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19 views

Trying to Understand Power Series Expansion of Cumulant Generating Function

I'm having difficulty expanding the cumulant generating function, $K_X(t)$ as a power series. Specifically, I'm trying to go from the definition of the cumulant generating function, i.e. $$K_X(t) = \...
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2answers
29 views

Linear Least square estimate of $x^3$ given $x$ and the moments

I have been struggling to find a direction on how to proceed with the following problem. Given that $x$ is a zero mean (non-Gaussian) random variable with moments E$(x^n)=\mu_n$. I need to find the ...
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Moments of the horseshoe prior?

Are the first two moments well defined for the horseshoe prior? I would say that the expectation is zero but the variance does not exist. Using the following argument. Let $$\beta_i \mid \lambda_i, \...
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1answer
94 views

Fourth order moments of the sum of multivariate normal distribution

Suppose $X\sim \mathcal{N}_d(\boldsymbol{0},\boldsymbol{\Sigma})$ follows a $d$-dimensional multivariate normal distribution. Let $\text{A}_i$ and $\text{A}_j$ be two arbitrary symmetric $d\times d$ ...
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Clarification on traditional notation for moments

$\newcommand{\E}{\mathrm{E}}$ I have been going over the Hill et al. (1976) paper on Johnson distributions and struggling with some notation that is undefined but I believe might have been considered ...
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How well is a power law distribution described by the first four moments?

For a normal distribution, the first two moments (mean and variance) are sufficient statistics for the entire distribution. Suppose I have a power law distribution, and I have data on the first, ...
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Is the posterior of a random variable's mean necessarily the mean of that random variable's posterior?

Let's say I have a model that's like, $$ Y \;|\; \theta_1 \sim P(Y \;|\; \theta_1) $$ $$\theta_1 \;|\; \theta_2 \sim P(\theta_1 \;|\; \theta_2) $$ $$ \theta_2 \;|\; \theta_3 \sim P(\theta_2 \;|\; \...
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Finding the distribution and its parameters with moment generating function

I am learning basic statistics and I am trying to solve the (example) problems but I can't figure out how to solve the following problem. I understand how to use the MGF to find expected value and ...
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112 views

Joint raw moments of multivariate normal

Let $X_1,X_2$ follow a bivariate standard normal distribution with some non-zero correlation coefficient, $\rho\neq 0$. Let the function $f(z) = z^k,\; k=1,...$. By Stein's lemma, we have that $$\...
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The lack of correlation determines the second-degree cross-moments (covariances) of a multivariate distribution?

It is given in the following image that lack of correlation determines the second-degree cross-moments (covariances) of a multivariate distribution,while in general statistical independence ...
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How is multivariate Gaussian distribution is determined by its second moments alone?

The following statement is given in Unsupervised Learning chapter of the book Elements of Statistical Learning. Since the multivariate Gaussian distribution is determined by its second moments ...
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Exponential Kth Moment Derivation

I have essentially a mathematical question, relating to deriving the formula for the kth moment of an exponential. I can't seem to work out how we get from the 2nd line to the 3rd line; i.e. the ...
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What do these sample moments represent? [duplicate]

For a given random variable $X$, the sample variance is (assuming large $n$ and so disregarding Bessel's correction, for simplicity): $$ \frac{\sum_i \left(x_i -\bar x \right)^2}{n} $$ For another ...
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Low-variance estimate for the mean of the sotfmax transformation of a variable

Consider a set of infintiely-differentiable convex functions real-valued functions $f_i: \mathcal X \rightarrow \mathbb R$, where $i$ varies from $1$ to $m$, and suppose we know all the moments of $...
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1answer
54 views

Inequality of first three moments for positive random variable

Let $X$ be a positive random variable (let's say finitely supported). Let $\mu_r = \mathbb{E} X^r$ be the $r$-th moment. Is it true that $$\mu_3 \geq \mu_1 \cdot \mu_2$$
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Expected value of (F(x)/x(1-F(x)) where F(x) is CDF?

I was solving a problem and encountered the following: \sum (f(x)/x) (F(x)/(1-F(x)) Here f(x) is pmf and F(x) is CDF. Is expected value of F(x)/x(1-F(x)) a known function? I found out that f(x)/(1-F(...
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1answer
126 views

Is there an analytical solution for cumulative probability for a distribution generated via a set of parameters? (mean, variance, skew, kurtosis) [closed]

I am trying to build a model that creates probabilistic estimates of the next location of an event, given a series of previous events (and other data). I want to maximize the likelihood that the next ...
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Express E[AAB] in terms of marginal moments and E[AB] only

For a pair of generic random variables $(A, B)$, I am trying to find a way to express $$E[A^2B]$$ using only the marginal moments $$E[A^k], E[B^k], k=1,2,\ldots $$ and the second joint moment $$...
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82 views

Method of Moment Estimator — Uniform Dist

Find the two method of moment estimators for $\theta$ given that $Y_i | \theta$ is distributed i.i.d U(0,$\theta$). We know that E($Y_1$) = $\frac{\theta}{2}$ and Var($Y_1$) = $\frac{\theta^2}{12}$ ....
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2answers
109 views

Fourth moment of unit-variance t-distribution

I am considering the unit-variance t-distribution. I have read that the fourth moment in such case is given by 3(v-2)/(v-4) where v is the degrees of freedom. Can someone explain how does this follow?...
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359 views

Generate random variable with given moments

I know first $N$ moments of some distribution. I also know that my distribution is continuous, unimodal and well shaped (it looks like gamma-distribution). Is it possible to: Using some algorithm, ...
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31 views

Partial quantile moments?

The integral from 0 to x* of x to the n times pdf(x) is usually called the nth partial moment function. Does the corresponding integral from 0 to y* of y to the n of the quantile density function, qdf(...
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1answer
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Expecations calculation question (regarding the autocovariance sequence of the square of a zero-mean stationary proces)

I have a zero-mean stationary (weak) process ${X_t}$ (meaning $\operatorname{cov}(X_t,X_{t+\tau})$ is a function of $|\tau|$ only for all $t$) and from it we get $Y_t$ such that $Y_t = X^2_t$ . In ...
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Log-measure of uncertainty

I would like to investigate if the following quantity has already a well-known meaning in statistic: $$G_{\eta}(X) =\frac{1}{\eta} \log\Biggl[1 + \lim_{N\to +\infty}\sum_{n=2}^N\frac{\eta^n}{n!}\...
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1answer
125 views

If higher order moment is finite, then lower order moment finite? [duplicate]

I'm trying to prove the theorem below. If $E(|X|^n)<\infty$ for some positive integer $n$, then $E(|X^k|)<\infty$ for every positive integer $k$ such that $k<n$ Here's what I've tried. ...