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

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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) = ...
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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 ...
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24 views

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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27 views

Cumulant and moment names beyond variance, skewness and kurtosis

In physics, starting from position $x(t)$, one obtains rates of changes via derivatives with respect to time: velocity, acceleration, jerk, jounce (4th order). Some have proposed snap, crackle, pop ...
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19 views

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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66 views

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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52 views

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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53 views

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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16 views

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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9 views

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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35 views

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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37 views

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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28 views

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 ...
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64 views

The connection between a random variable's moments and its tails

Suppose I know all the moments of some random variable XX. When is knowledge of the moments sufficient to give one an understanding of the tail behavior? Is there a nice way to show that a random ...
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130 views

Second moment method, Brownian motion?

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...
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29 views

First two moments of a quadratic form in which the vector and matrix are random (though independent)

$\DeclareMathOperator\tr{\mathrm{tr}}$Let $x$ be a standard normal $p$-variate random variable, which is independent of a symmetric positive definite random matrix $Y$. I would like to compute the ...
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11 views

Why does the moment generating function of a chi-squared random variable only exist for t<1/2?

I have found that for a chi-squared ($n$ degrees of freedom) random variable $X$, $M_X(t)= (1-2t)^{-n/2}$. I am told that this only exists for $t<1/2$. Why is this?
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9 views

Joint cumulants of Zn2 characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus ...
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Expectation of log determinant of complex Wishart random variables

Suppose $p$-dimensional complex positive matrix $X$ follows complex Wishart distribution whose density is given by $$ f(X) = ...
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300 views

How to fit an approximate PDF (i.e.: density estimation) using the first k (empirical) moments?

I have a situation where I am able to estimate (the first) $k$ moments of a data-set, and would like to use it to produce an estimation of the density function. I already came across the Pearson ...
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34 views

Better Moment Inequalities

How do you determine if a moment inequality is better than another? Say for example, compare the Chebyshev's inequality with this nameless inequality where P{|X|≥ Kσ} ≤ (μ4-σ4)/(μ4+K4σ4-2K2σ4). I ...
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improving accuracy for sample skewness and kurtosis under Student's t distribution with low degrees of freedom

There is no skewness in Student's t distribution. Given the degrees of freedom, the kurtosis estimator is given by $\frac{6}{\nu - 4}$. However, when sampling form a t-distribution with low degrees of ...
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Proving that the absolute moment of order a exists

Given $$f(x)= \frac{ka^k}{(x+a)^{k+1}}$$ with $x \geq 0, a > 0$. Show $E[|X|^a]< \infty$ for $a < k$. I am not really sure what theorems can be used to prove this. What I've been thinking ...
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30 views

Can any distribution be fully described by the [infinite] set of all its moments? $\{E[x^n]\}_{n\in N}$ [duplicate]

Is it possible describe any distribution uniquely by the infinite set of all it's moments $\{E[x^n]\}_{n\in N}$ If yes, does that include discrete, truncated, etc. distributions? If not, is this ...
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101 views

Convergence of moments of binomial to Poisson

As is well known, the $\mathsf{Binomial}(n,p)$ distribution converges to the $\mathsf{Poisson}(a)$ distribution as $n\rightarrow \infty$, $p\rightarrow 0$ with $np=a$. I'm pretty sure that the ...
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133 views

Undefined central moments; How to show that calculating a sample mean does not make sense?

Let $f$ be some probability density function with undefined central moments. For example, suppose $f$ is a Cauchy distribution. Say I draw two samples of size $N=100$ from that distribution. The mean ...
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47 views

Estimate betabinomial distribution parameters from weighted observations

Suppose we observe $N$ independent Bernoulli sequences of different lengths. Let $k_i$ be the number of trials in each observation, and $o_i$ be the number of "successes": ...
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Strictly Stationary Time Series with Infinite Moments

Can someone give me an example of a strictly stationary time series with infinite moments? I am reading a book on Time Series by Wayne A. Fuller where it is said that a strictly stationary time series ...
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How to reconstruct moments from aggregated data?

Let $X_{1\ldots n}$ be a stochastic variable that is log-normal distributed, with parameters $\mu$ and $\sigma$. Now suppose all $X_i$ are aggregated into $Y_{1\ldots m}$ where $Y_1$ is the mean of ...
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If the $(n-1)$th moment exists does the $n$th moment necessarily exist?

Let's suppose the distribution is unknown but that the second moment is known to be finite. Doesn't this imply that the distribution should fall off exponentially fast and therefore higher moments ...
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Shifted log-normal distribution and moments

I already know how to get the moments if $X$ is log-normally distributed. But what happens when $X$ is being shifted: $Y=aX+b$, $a>0$ and $b>0$. How to compute the moments of $Y$?
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Estimate variance from multiple distribution

Suppose I have $N$ feature vector and for each feature I have calculated the mean and variance. Now suppose I concatenate all the feature vector (appending one behind the other) and now if I want to ...
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Let $\mathbf{Y}$ be a random vector. Are $k$th moments of $\mathbf{Y}$ considered?

I am self-learning on linear model theory right now, and one thing I find surprising is that although $\mathbb{E}[\mathbf{Y}]$ is defined for a random vector $\mathbf{Y} = \begin{bmatrix} y_1 \\ y_2 ...
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Non-normal distributions with zero skewness and zero excess kurtosis?

Mostly theoretical question. Are there any examples of non-normal distributions that has first four moment equal to those of normal? Could they exist in theory?
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L-Kurtosis calculation

I am doing statistical analysis on some signal data and after some reading was thinking that L-kurtosis would be a good numerical value to use in differentiating delta trains and sine waves with ...
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3D Zernike moments vs. Spherical Harmonics. Which one has higher discriminative power as shape descriptor?

I am looking for a comprehensive study that has performed comparison of different 3D shape descriptors for classification/clustering problems. Particularly, I am interested in 3D Zernike moments vs. ...
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106 views

Sample variance from second moment

if I have a sample composed by values lesser than 1 and i want to compute the sample variance with $ \frac{n}{n-1}(\langle x_i^2 \rangle - \langle x_i \rangle^2)$ how can i do? Because the mean of the ...
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Looking for a distribution where: Mean=0, variance is variable, Skew=0 and kurtosis is variable

I am aiming to run simulations in order to estimate the influence of the distribution of $Y$ (independent variable) on a certain binary outcome $X$ (dependent variable). $Y$ must always has a mean of ...
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Tail of the inverse cdf

I am almost sure I have already seen the following result in statistics but I can't remember where. If $X$ is a positive random variable and $E(X)<\infty$ then $\epsilon F^{-1}(1-\epsilon) \to 0$ ...
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Why moments of expectation are known as “moments” [duplicate]

I am studying moments of expectation, and seen the formulas for computing the moments. There is one thing I am not clear of, and not getting answer for that. Why moments are named as moments? To my ...
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111 views

How to calculate the absolute central moment of a Binomial distribution?

There is an experiment. The coin is tossed $n$ times with $p = 0.5$. The experiment is repeated $k$ times. I need to calculate the average central moment. For example, let $n = 5$ and $k = 3$. $[0, ...
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Expectation Propagation - Computing mean and variance of error function

I'm still trying to wrap my head around computing the moments for the expectation propagation algorithm and whether I can use it for the following example: say i have a product of distributions which ...
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Moments and density tails

Assume that the first $n$ moments $m_1,\dots\,m_n$ of a random variable $X\in\mathbb{R}$ are known, but not its probability density function $p(x)$. Does there exist a methodology to characterize ...
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Constructing a continuous distribution to match m moments

Suppose I have a large sample drawn from a continuous distribution, size n, and 2 < m << n moments from that sample. Alternatively, suppose I have been given those moments by an angel, ...
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Determining approximate prediction interval using moments

Given the values of mean, and the next 3 central moments of a continuous random variable X > 0 with unknown pdf, is it possible to derive an approximate interval $(a,b)$ within which X will fall with ...
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222 views

Some basic questions related to the moments of a probability distribution

The available text books in most cases avoid greater details regarding some of the topics related to the moments of a probability distribution and I feel some of those topics are not really clear to ...
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104 views

Normal method of moments derivation explanation of Algebra step

In deriving normal estimators using method of moments, why does the below equality hold? $$ \frac{1}{n} \sum X_i^2 - \bar{X}^2 = \frac{1}{n} \sum (X_i - \bar{X})^2 $$ This is from Example 7.2.1 from ...
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32 views

Choosing among PDFs

This is a pretty broad question. I just learned that two random variables can have the same moments but different PDFs. Take $\mathbb{E}[X_i] = \mu$ and $\mathbb{Var}[X_i] = \sigma^2$. Since there are ...
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Transform sample to achieve target mean, skewness, etc

I have a sample of data with N values from which I calculate basic moments such as mean, standard deviation and skewness. I will then change these moments to different values according to my own ...