Tagged Questions

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

45 views

Calculate $E(X)$ and $\sigma(X)$ for a simple urn problem

In a box there are 4 red and 7 green balls. A random sample of 5 of the balls is made. Let the number of red balls in the sample be $X$ (a random variable). Calculate $E(X)$ and $\sigma(X)$
20 views

tensor cross covariance calculation

I have an $N \times P$ matrix $X$ (with $\Sigma$ variance covariance matrix $P \times P$) and I am using R to calculate the tensor product $E\{(X-\mu)(X-\mu)' \otimes (X-\mu)(X-\mu)'\}$ (tensor ...
74 views

How to prove three properties of the moment generating function? [duplicate]

The moment generating function of a random variable $X$ is defined to be the function $$M_{X}(t)=E(e^{tX})=\sum_{n=0}^{\infty}\frac{E(X^n)}{n!}t^n.$$ Let $I=\{t\in\mathbb R:M_{X}(t)<\infty\}.$ I ...
60 views

In finding the moment generating function why do we multiply by $e^{tx}$ for each pmf term?

The moment generating function that is associated with the discrete random variable $X$ and pmf $f(x)$ is defined as: $$M(t) = E\left[e^{tX}\right] = \sum_{x \in S} e^{tx} f(x).$$ Where does this ...
26 views

Computing non-central moments and normalizer of a quartic exponential distribution

Consider a random variable $X$ which has quartic exponential distribution: $$X \sim P(x)=\frac{1}{Z}e^{ax + bx^2 + cx^3 + dx^4}$$ How can one compute $Z$ or non-central moments $E X^k$ given that they ...
1k views

101 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, $\mathrm{E}(X)$ and $\mathrm{E}(X^2)$. What about values like $\mathrm{E}(X^{1.5})$ or $\mathrm{E}(X^{-1})$? Have they been investigated? ...
550 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 ...
55 views

How to show that $\mathrm{mgf}$ $M(s)$ and $\mathrm{pgf}$ $P(s)$ are related?

Let $X$ be an integer-valued $rv$ with $\mathrm{pgf}$ $P(s)$ (probability generating functions) and suppose that $\mathrm{mgf}$ $M(s)$ (moment generating functions) exist for $s∈(-s_0,s_0),s_0>0$. ...
1k 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 ...
94 views

Translating R code about treatment into Effect size. Expected mean and variance

Using R, I created groups of individuals with trait values. Then I simulated a treatment that modified their trait value (see below). Finally I run a one-way Anova on them using the individuals traits ...
22 views

How to improve location and scatter estimation conditioning on higher order statistics?

Using sample moments, how can the mean and variance estimators be improved if e.g. skewness and kurtosis are known exactly? And what about using estimates for these instead, which should imho be of no ...
323 views

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

Let $X_i$ be independent, normally distributed random variables, for $1\leq i\leq N$. What is the distribution of $Y_m=\frac1 N \sum_{i=1}^N X_i^m$? Every high school student knows part of the ...
496 views

172 views

Moments of the Kolmogorov distribution

Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.
53 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 ...
112 views

152 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 ...
111 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 ...
22 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 ...
217 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 ...
394 views

92 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), ...