Questions tagged [moments]
Moments are summaries of random variables' characteristics (e.g., location, scale). Use also for fractional moments.
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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?
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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 ...
- 113
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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 $Cov(X_1,X_2)$.
...
- 11
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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 ...
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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 ...
- 215
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15
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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 ...
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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 ...
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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? ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $n$
$$ E[X_n] < \...
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Bayesian VAR: Derivation of predictive distribution for reduced form VAR
I have a standard reduced form VAR of type without intercept: $y_t = A_{1}y_{t−1} + \ldots + A_py_{t−p} + e_t$, $e_t \sim N(0,Σ)$.
I need to derive the predictive posterior distribution $p(y_{T+h}|y_{...
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Are there maximum entropy distributions with fixed moments of a certain order?
A characterization of the multivariate Gaussian distribution with a fixed mean and covariance matrix is that it is the unique probability distribution that maximizes differential entropy. That is, the ...
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Central limit theorem : relaxing assumption of all finite moments
Consider $S_n = \sum_{i = 1}^n b_{i,n} X_{i,n}$ where
$X_{i,n}$ are random variable neither independent neither identically distribution and $b_{i,n}$ are weights satisfying the Lindeberg condition. I ...
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Is it true that $\langle X^4\rangle \ge 3 \langle X^2\rangle^2$?
Consider a real random variable $X$ with zero mean. Does the following inequality hold in general?
$$\langle X^4\rangle \ge 3 \langle X^2\rangle^2$$
I'm not sure how to prove this or if a counter-...
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How to apply the central limit theorem on higher order moments?
If I have a set of $N$ independent samples from a probability distribution $P(X)$, $X_i\sim P(X)$, then I know from the central limit theorem (assuming the distribution is well behaved) that the ...
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Convergence of moment of functional of random variable
Define $X_n$ a continuous random variable that converges in distribution to $X$. Morever, we know that $E[|X_n|^p] \rightarrow E[|X|^p]$ for some $p > 0$.
Then, could we prove that for any ...
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Derivative conditional moments binary random variable
Let $e$ be a continuously distributed RV with pdf $f$ and let $q( x )$ be a binary RV that depends on the former through the relation $q ( x ) = 1[h( x ) \geq e ]$, where $h$ is a well-behaved ...
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Calculating the n-th moment of a RV, including negative fractional moments
I am stuck trying to solve the following exercise..
Let $X: \Omega\to [a,b] \subset \mathbb R$ be a uniformly distributed random variable. Compute the n-th moment of $X$, i.e. compute $\mathbb E[X^n]$ ...
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Absolute third central moment for standard distributions reference
I have to write an R function that computes the absolute third central moment (i.e. $\mathbb{E}[|X-\mathbb{E}[X]|^3]$) in the cases that you are given the name of ...
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Possible typo in discussion of moments of a random variable
I'm struggling to understand some notation in this excerpt from Larsen & Marx. Under "Comment" j is defined as ...
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What is the correct Gaussian to use given that I know the true second moment and have one sample from the distribution?
I have some random variable $X$ that is normal distributed. I know the second moment $s = \mathbb{E}[X^2]$ and have one sample x from the distribution. How can I get an unbiased estimate of the mean ...
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Motivation for using moment about mean versus moment about the origin
When calculating moments of probability distributions, what would motivate you to take the moment about the mean (called "central moment") versus the standard moment about the origin?
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Find nth central moment for exponential distribution
I am trying to figure out what the nth central moment is for the exponential distribution.
Here is the formula for the nth moment:
$$
\mathop{\mathbb{E}}{[x^n]} =\dfrac{n!}{\lambda^n}
$$
My question: ...
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How should I estimate the variance in a sample or population from the sample range?
Suppose I wish to know the variance within a sample or of the population from which it is drawn. However, I do not have true measurements for most of my "observations". Think of them as like ...
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Estimate Conditonal Moments from Conditonal Quantiles
In Chang et al. "The Higher Moments of Future Earnings" (2014), the authors say say that based on (predicted) conditonal quantiles of a variable $y$, one can derive the (predicted) ...
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When would correlation between two variables not exist?
If we have two random variables $X$ and $Y$, then $\text{corr}(X,Y)=\dfrac{
\text{cov}(X,Y)
}{
\sqrt{
\text{var}(X)\text{var}(Y)
}
}$.
This correlation will not be defined if either variable has an ...
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Expected squared dot product between IID Gaussian vectors?
Suppose $x,y$ are IID samples from a Gaussian distribution in $\mathbb{R}^d$. The following seems true:
$$2\ \mathbb{E}\left[\langle x, y\rangle^2\right] = \mathbb{E}\left[\|x\|^4\right]-\mathbb{E}\...
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What is $E\left[\frac{X^2}{X^2+Y^2}\right]$ if $X$ and $Y$ are normally distributed but not iid.?
I assume that X and Y are normally distributed with individual mean and variance. So far, I have found that an analytic expression exists for $E[X^2+Y^2]$, $E[X^2*Y^2]$ and $E[X^2*(X^2+Y^2)]$, all ...
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If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than P(Y>x)?
$X$ and $Y$ are two non-negative continuous random variables. The moments of $X$ are $\mu_i$ while that of $Y$ are $\nu_i$.
We know that $\mu_1=\nu_1$ and $\mu_i \ge \nu_i$ for $i=2,3,\ldots$
Can one ...
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When are the moments of the square root of a random variable less than the moments of a random variable?
Let $X$ be a non-negative real-valued random variable. Can one find conditions for when the moments of $\sqrt{X}$ are less or equal to the moments of $X$? A specific example would be the chi-square ...
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Moments of inverse of a non-central chi distributed variable
I have a non-central chi variable $r$ with the distribution,
\begin{align}
p(r) = \frac{r^3\lambda}{(\lambda r)^{3/2}}\exp\left[-0.5(r^2 + \lambda^2)\right]I_{1/2}(r\lambda)
\end{align}
I'm looking ...
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Central cross moments of independent random varibales
The central cross-moments of a random vector $[X_1, X_2,\dots, X_n]$ is
$$ \mu_X(n_1,n_2,\dots,n_n) =E[(X_1-E[X_1])^{n_1}(X_2-E[X_2])^{n_2}\dots(X_n-E[X_n])^{n_n}]$$
I wish to know if $\mu_X(1,1,\dots,...
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Transformation function on random variable to match desired CV and Skewness
Let's say I have a continuous random variable (X) that takes any value greater than 0 and with certain Mean, CV, and Skewness values. The data can follow any distribution. I want to apply a ...
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Differential entropy / variance
So I‘m basically more or less trying to prove what is stated in the answer by syeh_106 here:
Is differential entropy always less than infinity?
if the variance is finite, then the differential ...
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If the variance of $X^2$ is zero, when is the variance of $X$ not zero
$X$ is a random variable with mean $\mu$, variance $\sigma^2$, skewness $\gamma$ and kurtosis $\kappa$, and
$ \mathtt{Var}[X^2]=0.$
Under what conditions can $\mathtt{Var}[X]=\sigma^2 >0$?
Is my ...
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Kurtosis is greater or equal to square of skewness plus one [duplicate]
Given is a random variable $X$ with finite fourth moment. Let $\gamma_3$ and $\gamma_4$ denote its skewness and kurtosis respectively.
I want to prove that $$\gamma_4\geq 1+ \gamma_3^2$$
I have seen a ...
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How to empirically calculate the first 4 moments of a stochastic process from a simulated array of data?
I am a little bit confused about empirically calculating the first 4 moments of a stochastic process. I have an array of values that correspond to a stochastic process. That is, $[X_1,X_2,...,X_n], ...
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Provide an example of a dataset where maximum likelihood is inapplicable as third moments and fourth moments "assumptions" do not apply
An additional complication arises with estimation, since maximum likelihood estimation may not be feasible without making unrealistically strong ?????"assumptions"????? about third‐ and ...
user318514
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How can I standardize a sample to fixed skewness and kurtosis? [closed]
Standardizing a sample of a random variable to mean 0 and standard deviation 1 are common practice. However, I would like to also standardize its skewness to 0 and its kurtosis to 3, but preserve ...
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Gaussian fourth-moment formulas?
Suppose $X$ is an $m\times n$ random matrix where rows are I.I.D. samples of $n$-dimensional Gaussian, and $A$ is an $n\times n$ matrix. I'm looking for the value of the following quantities:
$$E[X^T ...
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Are moments more robust than MLE?
I am an MBA Student taking courses in Statistics.
We are learning about different ways to estimate the parameters (i.e. coefficients) of a Regression Model. Our professor indicated that there are two ...
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What does Cayley's hyperdeterminant of a 2x2x2 mixed-product moment tensor tell us about how two variables are related?
Suppose we have a collection of random variables $S = \{ X_0, X_1 \}$ encoded into the $2 \times 2 \times 2$ tensor
$$\mathcal{C}[i, j, k] = \mathbb{E}[X_i X_j X_k]$$
where $X_i, X_j, X_k \in S$ and $...
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Does $E\frac{1}{\|x\|^{4}} \rightarrow \frac{1}{E\|x\|^4}$ in high dimensions?
Suppose we have a Gaussian distribution centered at zero, covariance matrix $\Sigma$ with $\operatorname{Tr}\Sigma=1$ and $\operatorname{Tr}\Sigma^2=\frac{1}{2}$
When I try various sequences of such ...
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Does k moments imply k + $\epsilon$ moments?
If you have a random variable $X$ such that $\mathbb{E}(|X|^k) < \infty$, does it follow that $\mathbb{E}(|X|^{k+\epsilon}) < \infty$ for some (potentially small) $\epsilon > 0$? If not, ...
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Unbiasing estimator of $\|\Sigma\|_F^2$
I have access to samples of some distribution with second-moment matrix $\Sigma=E[xx^T]$ and need an estimate of $\|\Sigma\|_F^2$ (which can be used to set optimal size for LMS)
We can use Frobenius ...
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Analytical Expression for Moment of Generalized chi-squared distribution
Consider $Z\sim N(0,1)$ and the moments:
$$
E\left[ \left(C_1(Z+\sqrt{\lambda})^2 - C_2\right)^t\right].
$$
Here, $C_1$, $\lambda$ and $C_2$ are arbitrary constants which are all positive. $t$ is a ...
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How to compute this moment of a bivariate normal distribution?
Consider that $X \sim N(\mu_X, \sigma_X^2)$ and $Y \sim N(\mu_Y, \sigma^2_Y)$ and $\text{Cov}[X, Y] = \sigma_{XY}$ where $\sigma_{XY} \ne 0$.
How can an expression for $\text{E}[Y(X-\mu_X)^p]$ in ...
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