# Tagged Questions

For questions about the moments of a random variable, that is, the expectation of $X^k$ where $X$ is a random variable and $k$ an integer (or a real number with $X$ non-negative).

2answers
786 views

### What's so 'moment' about 'moments' of a probability distribution?

I KNOW what moments are and how to calculate them and how to use the moment generating function for getting higher order moments. Yes, I know the math. Now that I need to get my statistics knowledge ...
3answers
958 views

### Moments of a distribution - any use for partial or higher moments?

It is usual to use second, third and fourth moments of a distribution to describe certain properties. Do partial moments or moments higher than the fourth describe any useful properties of a ...
1answer
325 views

### Intuition for higher moments in circular statistics

In circular statistics, the expectation value of a random variable $Z$ with values on the circle $S$ is defined as $$m_1(Z)=\int_S z P^Z(\theta)\textrm{d}\theta$$ (see wikipedia). This is a very ...
1answer
467 views

### Error in normal approximation to a uniform sum distribution

One naive method for approximating a normal distribution is to add together perhaps $100$ IID random variables uniformly distributed on $[0,1]$, then recenter and rescale, relying on the Central Limit ...
3answers
150 views

### Approximating $Pr[n \leq X \leq m]$ for a discrete distribution

What's the best way to approximate $Pr[n \leq X \leq m]$ for two given integers $m,n$ when you know the mean $\mu$, variance $\sigma^2$, skewness $\gamma_1$ and excess kurtosis $\gamma_2$ of a ...
4answers
1k views

### A transform to change skew without affecting kurtosis?

I am curious if there is a transform which alters the skew of a random variable without affecting the kurtosis. This would be analogous to how an affine transform of a RV affects the mean and ...
1answer
1k views

### Existence of the moment generating function and variance

Can a distribution with finite mean and infinite variance have a moment generating function? What about a distribution with finite mean and finite variance but infinite higher moments?
2answers
440 views

### A proof involving properties of moment generating functions

Wackerly et al's text states this theorem "Let $m_x(t)$ and $m_y(t)$ denote the moment-generating functions of random variables X and Y, respectively. If both moment-generating functions exist and ...
1answer
286 views

### Central Moments of Symmetric Distributions

I am trying to show that the central moment of a symmetric distribution: $${\bf f}_x{\bf (a+x)} = {\bf f}_x{\bf(a-x)}$$ is zero for odd numbers. So for instance the third central moment {\bf ...
2answers
878 views

### Exponential weighted moving skewness/kurtosis

There are well-known on-line formulas for computing exponentially weighted moving averages and standard deviations of a process $(x_n)_{n=0,1,2,\dots}$. For the mean, \$\mu_n = (1-\alpha) \mu_{n-1} + ...
1answer
599 views

### Compute approximate quantiles for a stream of integers using moments?

migrated from math.stackexchange. I'm processing a long stream of integers and am considering tracking a few moments in order to be able to approximately compute various percentiles for the stream ...
1answer
430 views

### Testing two independent samples for null of same skew?

What tests are available for testing two independent samples for the null hypothesis that they come from populations with the same skew? There is a classical 1-sample test for whether the skew equals ...
0answers
192 views

### Physical/pictoral interpretation of higher-order moments

I'm preparing a presentation about parallel statistics. I plan to illustrate the formulas for distributed computation of the mean and variance with examples involving center of gravity and moment of ...
1answer
85 views

### Eighth order moment

I read Nonlinear Dimensionality Reduction by Lee and Verleysen [Google Books] and came across the following theorem (p. 8): Let $\mathbf{y}$ be a $D$-dimensional vector \$[y_1, \ldots, y_d, ...
1answer
254 views

### Proof that if higher moment exists then lower moment also exists

The $r$-th moment of a random variable $X$ is finite if $$\mathbb E(|X^r|)< \infty$$ I am trying to show that for any positive integer $s<r$, then the $s$-th moment $\mathbb E[|X^s|]$ is ...
2answers
457 views

### Mean and variance of log-binomial distribution

If X is a random variable with a normal distribution, then Y = exp(X) has a log-normal distribution. Likewise if X is a random variable with a binomial distribution, then Y = exp(X) has a ...
2answers
279 views

### Determine whether a n-th finite moment of X exists

I have a question which asks: Determine those values of the positive integer n for which a finite nth moment of X about zero exists. How should I approach this question? Does it depend on the ...
2answers
271 views

### How to understand moments for a random variable?

Wikipedia says that the name of concept comes from physics, but I cannot find any similarity between these two concepts.
1answer
129 views

### Mixture distributions moments if one distribution has undefined/infinite moments

Consider probability density functions $f_{1}\left(x\right)$ and $f_{2}\left(x\right)$ and the mixture distribution f_{3}\left(x\right)\equiv ...
1answer
364 views

### Link between moment-generating function and characteristic function

I am trying to understand the link between the moment-generating function and characteristic function. The moment-generating function is defined as:  M_X(t) = E(\exp(tX)) = 1 + \frac{t E(X)}{1} + ...
1answer
62 views

### nth moment, for 0 < n < 1 or n <0, do they exist?

I am interested in the moments, we have for instance the mean, E(X) and E(X^2). what about values like $E(X^{1.5})$ or $E(X^{-1})$? Have they been investigated?
1answer
530 views

### Max Entropy Solver in R

I am trying to solve for the parameters in the maximum entropy problem in R using the nonlinear system: $\ \int_l^u e^{a+bx+cx^2}dx=1$ $\ \int_l^u x e^{a+bx+cx^2}dx=\mu$ \$\ \int_l^u x^2 ...
1answer
86 views

### covariance of RVs under a nonlinear transformation

I have a multivariately distributed random 3-vector ...
1answer
288 views

### One sided Chebyshev inequality for higher moment

Is there an analogue to the higher moment Chebyshev's inequalities in the one sided case? The Chebyshev-Cantelli inequality only seem to work for the variance, whereas Chebyshevs' inequality can ...
1answer
77 views

### Using MGF for multivariate random variables

How do you use MGF for solving moment based questions for multivariate random variables? For the single variable case, we: find $E(e^{tX})$, find the interval in which it exists (around 0), ...
1answer
188 views

### Central moments of a gaussian mixture density?

Given the pdf $f(x) = \sum_i \omega_i \mathcal{N}(x; \mu_i, C_i )$ of a gaussian mixture density, where the $i$-th component has mean $\mu_i$ and covariance matrix $C_i$ and the weights $\omega_i$ sum ...
2answers
130 views

### Moments of the Kolmogorov distribution

Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.
1answer
69 views

### Does finite kth moment imply lesser moments are finite? [duplicate]

Possible Duplicate: Proof that if higher moment exists then lower moment also exists For a random variable $X$, lets say I know $E[X^k]$ is finite and I know that $E[X]$ is finite. Can I ...
1answer
76 views

### Question about inverse in a two-step estimator as a joint GMM-estimators approach

I'm reading Newey & McFadden - Large sample estimation and hypothesis testing (in the Handbook of Econometrics, Volume 4, 1994, page 2178). My model which I'm interested in has some former ...
1answer
2k views

### Finding the Moment Generating Function of Chi-Squared Dist

I'm tasked with finding the MGF of a $\chi^2$ random variable. I think the way to do is is by using the fact that $\Sigma_{j=1}^{m} Z^2_j$ is a $\chi^2$ R.V. and that MGF of a sum is the product of ...
1answer
94 views

### Combining two covariance matrices

I'm calculating the covariance of a distribution in parallel and I need to combine the distributed results into on singular Gaussian. How do I combine the two? Linearly interpolating between the two ...
1answer
1k views

### Kurtosis/4th central moment in terms of mean and variance

Is it possible to express the kurtosis $\kappa$, or the 4th central moment $\mu_4$, of a random variable $X$ in terms of its mean $\mu = E(X)$ and variance $\sigma^2 = Var(X)$ only, without having to ...
1answer
224 views

### Moment generating function of the inner product of two gaussian random vectors

Can anybody please suggest how I can compute the moment generating function of the inner product of two gaussian random vectors, each distributed as $\mathcal N(0,\sigma^2)$, independent of each ...
2answers
257 views

### What is coskewness and how can it be calculated?

I would like to calculate coskewness of two random variables. However I couldn't find even basic information on this matter. Is there a standard definition? How to calculate it? If not what are my ...
1answer
80 views

### Difference between the two normal distributions

I have two random variables $X$ and $Y$ which follows Normal distribution , whose pdf's are given by \$f(x)= \frac{1}{2 \sqrt{2 \pi} \sigma}[e^{\frac{-(x-1)^2}{2 \sigma^2}}+e^{\frac{-(x+1)^2}{2 ...
1answer
135 views

### Advantage of central moment over moment?

Is there any advantage of using "central moments" over "moments" when approximating a distribution to a known distribution using moment matching? I have noticed that in lot of papers.
2answers
89 views

### Higher Moments of an Unknown Density Function

Given a R.V $X$ and that $\ E(X) = 0$ and $\ E(X^2) = \sigma^2$. Is there anyway to compute $\ E(X^3)$ without knowing the density function of $X$?
1answer
247 views

### What are the sampling distributions of higher moments of the normal distribution?

Let Xi be independent, normally distributed random variables, for 1≤i≤N. What is the distribution of Ym=N-1Σ(Xi)m? Every high school student knows part of the answer. The mean of Ym is ...
1answer
115 views

### Estimate the second moment of a latent variable using a conditionally unbiased proxy

The Setup: Let $X_t$ denote an unobservable stochastic sequence where the first two unconditional moments are finite constants; ie $\mathbb{E} X_t = \mu < \infty$ and \$\mathbb{E} X_t^2 = \gamma ...
1answer
136 views

### Question about a derivative of the 2nd-step moments in a two-step estimator as a joint GMM-estimators approach

I'm reading Newey & McFadden - Large sample estimation and hypothesis testing (in the Handbook of Econometrics, Volume 4, 1994, page 2176). In the model I'm interestend in has some former ...
1answer
133 views

### Identity of moment-generating functions

Are there any non-identical distributions which happen to have the same moment-generating function?
0answers
271 views

### Moments of Laplace distribution

I am a newbie in stat. I am working on the Laplace distribution for my algorithm. Could tell me the first what the four moments of the Laplace distribution are? Does it have infinite tail like the ...
2answers
93 views

### Do skewness and kurtosis uniquely determine type of distribution?

Inspired by this answer, I have following question: Is it enough to know just skewness and kurtosis in order to determine distribution that data comes from? Is there any theorem that implies this? ...
1answer
52 views

### 2-dimensional moment and rotation

Is it possible to get a simple formula linking a central moment to the same moment in a rotated frame, such as the relation between the central moment and the moment about the origin? The formula I ...
2answers
240 views

### Need an example of RV with a mean and no second moment

An example like the t-distribution with 2 degrees of freedom would not suffice as the second moment exists but equals inf.
1answer
46 views

### Finding moments for a theoretical density function

I am working on finding higher order moments for a given theoretical function, to be used in modelling of daily log-returns. The PDF is, $f_r(x) =$ \$\begin{cases} \quad ...
1answer
98 views

### Proving that MGF determines PDF when the PDF is defined for whole real line

If two PDFs have the same moment generating function that converges in an open set around 0, then the PDFs are same. This is a well known fact, but I can't find its proof. If the PDFs are defined ...
1answer
514 views

### How can I calculate central moments of a joint pdf?

Let's say I have two signals $x_1$ and $x_2$, each having $N$ samples, i.e.: $$x_1 = \{ x_{11}, x_{12}, ..., x_{1N} \}$$ $$x_2 = \{ x_{21}, x_{22}, ..., x_{2N} \}$$ The signals are both ...
1answer
194 views

### Variation of Skewness and Excess Kurtosis

I apologize ahead of time if this a trivial question. I am building probability models for games - and in order to test the game code, I am comparing the results of large iterations of games with the ...
0answers
82 views

### Multivariate normal - conditioning on absolute values

I’m reading a paper and really struggling with one appendix. Basically they derive conditional expectation of a multivariate normal, conditioning on absolute values. Let \boldsymbol y = ...
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