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).
4
votes
1answer
79 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 ...
0
votes
0answers
23 views
Distribution with fifth order moment?
I understand that the fifth moment of a distribution gives finer control of the asymmetry of the tails.
Please can you give me a reference to a distribution that can handle 5 moments (such as the ...
1
vote
2answers
96 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? ...
2
votes
1answer
232 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 ...
1
vote
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 ...
1
vote
0answers
31 views
Is it possible to calculate mutual information by moments generating functions?
I went to listen to a workshop and some audience asked the presenter how the moments can improve the mutual information. I am learning the MI(Mutual Information) and moments so don't have enough ...
4
votes
1answer
369 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} + ...
3
votes
2answers
130 views
Moments of the Kolmogorov distribution
Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.
0
votes
0answers
39 views
Confusion about using moment condition in a multiple regression model
The very simple case assumes that we have a model like $y = a + bx + e$ where the condition $cov(x,e)=0$ is true. Hence one can use the relationship of the moment conditions to estimate the parameter ...
2
votes
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 ...
3
votes
1answer
97 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 ...
2
votes
2answers
279 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 ...
4
votes
2answers
462 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 ...
3
votes
1answer
88 views
covariance of RVs under a nonlinear transformation
I have a multivariately distributed random 3-vector
...
2
votes
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 ...
2
votes
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 ...
0
votes
1answer
74 views
Proving that central moment is finite
I'm having trouble showing that the 2nd central moment is finite. I have $X_1,\ldots,X_n \overset{iid}{\sim} f(x)$ with $E[X_1]=\mu$ and $E[X_1^k]$ exists and is finite for any integer $k \geq 1$.
I ...
0
votes
0answers
15 views
Test dataset to assess validity of software implementation
I am writing a a=software implementation that computes arbitrary-order central moments. The implementation looks good, but I want to make sure I made no mistake.
Is there out there classical datasets ...
3
votes
1answer
70 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 ...
6
votes
1answer
290 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 ...
3
votes
1answer
77 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 ...
0
votes
1answer
90 views
Distribution function
Find (without using MGF) the mean and variance.
$$f(x) = \exp(-kx)x^{(r-1)}k^r/(r-1)!\ \text{ for }\ x>=0$$
$$f(x) = 0\ \text{ for }\ x<0$$
$r$ positive integer, $k>0$
1
vote
1answer
99 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 ...
0
votes
0answers
65 views
Working with an arbitrary number of sample moments
The $n^{th}$ moment of a distribution can be estimated from a vector of samples $(x_1,x_2,...x_k)$ by:
$$
\sum_{i=1}^{k} x_i^n
$$
Now, let's say I've calculated the first $m$ moments for my ...
2
votes
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 $X$?
4
votes
1answer
64 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?
1
vote
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
=
...
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), ...
5
votes
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, ...
7
votes
2answers
441 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 ...
2
votes
1answer
133 views
Identity of moment-generating functions
Are there any non-identical distributions which happen to have the same moment-generating function?
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?
3
votes
1answer
191 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 ...
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 ...
4
votes
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 ...
1
vote
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 ...
2
votes
1answer
136 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.
9
votes
1answer
474 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 ...
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 ...
5
votes
1answer
255 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 ...
1
vote
1answer
515 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 ...
1
vote
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 ...
1
vote
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.
3
votes
1answer
532 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
296 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 ...
19
votes
2answers
794 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 ...
1
vote
2answers
443 views
If the n(th) moment exists does it mean all smaller moments exist too?
I would like to prove the following statement:
If the $r$th moment of a random variable $X$ exists and is finite,
then all moments $1$ to $r-1$ exist and are finite.
Edit: I mean the raw ...
2
votes
1answer
249 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 ...
0
votes
0answers
175 views
Moments of function of Poisson process
Mirror thread on Mathoverflow.
(I'm new to Poisson processes, so please edit if my terminology is incorrect.)
This is a special case of a problem I'm working on; hoping for intuition that will ...
6
votes
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 ...
