# Questions tagged [moments]

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

571 questions
Filter by
Sorted by
Tagged with
3 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?
36 views

### 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 ...
1 vote
23 views

### 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 $Cov(X_1,X_2)$. ...
45 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 ...
17 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 ...
15 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 ...
21 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 ...
100 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? ...
1 vote
22 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 ...
62 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 ...
113 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 ...
53 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 ...
1 vote
26 views

55 views

### 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 ...
419 views

### 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 ...
1 vote
50 views

### 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 ...
163 views

### 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 ...
29 views

1 vote
35 views

### 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 ... 1 vote
164 views

### 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 ...
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: $$E[X^T ... 2 votes 1 answer 312 views ### Are moments more robust than MLE? I am an MBA Student taking courses in Statistics. We are learning about different ways to estimate the parameters (i.e. coefficients) of a Regression Model. Our professor indicated that there are two ... 1 vote 2 answers 67 views ### What does Cayley's hyperdeterminant of a 2x2x2 mixed-product moment tensor tell us about how two variables are related? Suppose we have a collection of random variables S = \{ X_0, X_1 \} encoded into the 2 \times 2 \times 2 tensor$$\mathcal{C}[i, j, k] = \mathbb{E}[X_i X_j X_k]$$where X_i, X_j, X_k \in S and ... 8 votes 1 answer 266 views ### Does E\frac{1}{\|x\|^{4}} \rightarrow \frac{1}{E\|x\|^4} in high dimensions? Suppose we have a Gaussian distribution centered at zero, covariance matrix \Sigma with \operatorname{Tr}\Sigma=1 and \operatorname{Tr}\Sigma^2=\frac{1}{2} When I try various sequences of such ... 2 votes 1 answer 57 views ### Does k moments imply k + \epsilon moments? If you have a random variable X such that \mathbb{E}(|X|^k) < \infty, does it follow that \mathbb{E}(|X|^{k+\epsilon}) < \infty for some (potentially small) \epsilon > 0? If not, ... 4 votes 0 answers 146 views ### Unbiasing estimator of \|\Sigma\|_F^2 I have access to samples of some distribution with second-moment matrix \Sigma=E[xx^T] and need an estimate of \|\Sigma\|_F^2 (which can be used to set optimal size for LMS) We can use Frobenius ... 0 votes 1 answer 60 views ### Analytical Expression for Moment of Generalized chi-squared distribution Consider Z\sim N(0,1) and the moments:$$ 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 ...
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 ...