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).
19
votes
2answers
829 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 ...
11
votes
3answers
990 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 ...
10
votes
1answer
334 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 ...
9
votes
1answer
493 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 ...
9
votes
3answers
151 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 ...
9
votes
3answers
2k 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 ...
8
votes
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?
7
votes
2answers
461 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 ...
6
votes
1answer
298 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 $${\bf ...
6
votes
2answers
912 views
Exponential weighted moving skewness/kurtosis
There are well-known on-line formulas for computing exponentially weighted moving averages and standard deviations of a process $(x_n)_{n=0,1,2,\dots}$. For the mean,
$\mu_n = (1-\alpha) \mu_{n-1} + ...
6
votes
1answer
652 views
Compute approximate quantiles for a stream of integers using moments?
migrated from math.stackexchange.
I'm processing a long stream of integers and am considering tracking a few moments in order to be able to approximately compute various percentiles for the stream ...
6
votes
1answer
450 views
Testing two independent samples for null of same skew?
What tests are available for testing two independent samples for the null hypothesis that they come from populations with the same skew? There is a classical 1-sample test for whether the skew equals ...
6
votes
0answers
196 views
Physical/pictoral interpretation of higher-order moments
I'm preparing a presentation about parallel statistics. I plan to illustrate the formulas for distributed computation of the mean and variance with examples involving center of gravity and moment of ...
5
votes
1answer
88 views
Eighth order moment
I read Nonlinear Dimensionality Reduction by Lee and Verleysen [Google Books] and came across the following theorem (p. 8):
Let $\mathbf{y}$ be a $D$-dimensional vector $[y_1, \ldots, y_d, ...
5
votes
1answer
260 views
Proof that if higher moment exists then lower moment also exists
The $r$-th moment of a random variable $X$ is finite if
$$
\mathbb E(|X^r|)< \infty
$$
I am trying to show that for any positive integer $s<r$, then the
$s$-th moment $\mathbb E[|X^s|]$ is ...
4
votes
2answers
486 views
Mean and variance of log-binomial distribution
If X is a random variable with a normal distribution, then Y = exp(X) has a log-normal distribution.
Likewise if X is a random variable with a binomial distribution, then Y = exp(X) has a ...
4
votes
2answers
283 views
Determine whether a n-th finite moment of X exists
I have a question which asks:
Determine those values of the positive integer n for which a finite nth moment of X about zero exists.
How should I approach this question? Does it depend on the ...
4
votes
1answer
99 views
Confidence error bars and “central point”: Should we emphasize the median?
Say I want to plot summary data with a point and a 95% confidence interval around that point. What should my point really be? Mean, mode, or median?
I know that mean = median for any symmetrical ...
4
votes
2answers
271 views
How to understand moments for a random variable?
Wikipedia says that the name of concept comes from physics, but I cannot find any similarity between these two concepts.
4
votes
1answer
131 views
Mixture distributions moments if one distribution has undefined/infinite moments
Consider probability density functions $f_{1}\left(x\right)$ and $f_{2}\left(x\right)$ and the mixture distribution
$$f_{3}\left(x\right)\equiv ...
4
votes
1answer
145 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 ...
4
votes
1answer
390 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} + ...
4
votes
1answer
66 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?
3
votes
1answer
562 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 ...
3
votes
1answer
103 views
covariance of RVs under a nonlinear transformation
I have a multivariately distributed random 3-vector
...
3
votes
1answer
140 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.
3
votes
1answer
319 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 ...
3
votes
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), ...
3
votes
1answer
193 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 ...
3
votes
2answers
136 views
Moments of the Kolmogorov distribution
Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.
3
votes
1answer
75 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 ...
3
votes
1answer
78 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 ...
3
votes
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 ...
2
votes
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 ...
2
votes
1answer
262 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 ...
2
votes
2answers
360 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 ...
2
votes
1answer
91 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 ...
2
votes
1answer
30 views
Moment generating function of multinomial distribution
How would one find the moment generating function of the multinomial distribution, $\underline{X} \sim \mathrm{multinomial}(n, \underline{p})$? I know that by definition we have $$M_X ...
2
votes
2answers
92 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 $X$?
2
votes
1answer
252 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 ...
2
votes
1answer
118 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 ...
2
votes
1answer
138 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 ...
2
votes
1answer
137 views
Identity of moment-generating functions
Are there any non-identical distributions which happen to have the same moment-generating function?
2
votes
0answers
277 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 ...
1
vote
2answers
108 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? ...
1
vote
1answer
53 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 ...
1
vote
2answers
242 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.
1
vote
1answer
48 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 ...
1
vote
1answer
102 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 ...
1
vote
1answer
526 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 ...
