Statistical Analysis Stack Exchange
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Questions tagged [moment-generating-function]

A description of a probability distribution which is related to the Laplace transform. Use also for its logarithm, the cumulant generating function.

https://en.wikipedia.org/wiki/Moment-generating_function

https://en.wikipedia.org/wiki/Cumulant

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How does saddlepoint approximation work?

How does saddlepoint approximation work? What sort of problem is it good for? (Feel free to use a particular example or examples by way of illustration) Are there any drawbacks, difficulties, things to watch out for, or traps for the unwary?
Glen_b
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Are CDFs more fundamental than PDFs?

My stat prof basically said, if given one of the following three, you can find the other two: Cumulative distribution function Moment Generating Function Probability Density Function But my econometrics professor said CDFs are more fundamental…
Stan Shunpike
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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?
Mgf
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Probability inequalities

I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an exponential upper bound over the probability that the sum of…
Farzad
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Proof that moment generating functions uniquely determine probability distributions

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 $m_x(t) = m_y(t)$ for all values of t, then X and Y…
Chris Simokat
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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} + \frac{t^2 E(X^2)}{2!} + \dots + \frac{t^n…
Giuseppe
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How would you explain Moment Generating Function(MGF) in layman's terms?

What is a Moment Generating Function (MGF)? Can you explain it in layman's terms and along with a simple & easy example? Please, limit using formal math notations as far as possible.
user366312
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Whether distributions with the same moments are identical

Following are similar to but different from previous posts here and here Given two distributions which admit moments of all orders, if all the moments of two distributions are the same, then are they identical distributions a.e.? Given two…
Tim
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Distribution of $\frac{\sum_{i=1}^n X_iY_i}{\sum_{i=1}^n X_i^2}$ where $X_i,Y_i$s are i.i.d Normal variables

Suppose $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ are i.i.d $\mathcal N(0,1)$ random variables. I am interested in the distribution of $$U=\frac{\sum_{i=1}^n X_iY_i}{\sum_{i=1}^n X_i^2}$$ I define $$Z=\frac{\sum_{i=1}^n X_iY_i}{\sqrt{\sum_{i=1}^n…
StubbornAtom
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What is the difference between moment generating function and probability generating function?

I am confused between the two terms " probability generating function" and "moment generating function." How do those terms differ?
manashi
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Bound on moment generating function

This question arises from the one asked here about a bound on moment generating functions (MGFs). Suppose $X$ is a bounded zero-mean random variable taking on values in $[-\sigma, \sigma]$ and let $G(t) = E[e^{tX}]$ be its MGF. From a bound used in…
Dilip Sarwate
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Distribution with $n$th cumulant given by $\frac 1 n$?

Is there any information out there about the distribution whose $n$th cumulant is given by $\frac 1 n$? The cumulant-generating function is of the form $$ \kappa(t) = \int_0 ^ 1 \frac{e^{tx} - 1}{x} \ dx. $$ I've run across it as the limiting…
guy
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Identity of moment-generating functions

Are there any non-identical distributions which happen to have the same moment-generating function?
user9437
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Necessary and sufficient condition on joint MGF for independence

Suppose I have a joint moment generating function $M_{X,Y}(s,t)$ for a joint distribution with CDF $F_{X,Y}(x,y)$. Is $M_{X,Y}(s,t)=M_{X,Y}(s,0)⋅M_{X,Y}(0,t)$ both a necessary and sufficient condition for independence of $X$ and $Y$? I checked a…
Silverfish
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Does there exist any univariate distribution that we can't sample from?

We have great variety of methods for random generation from univariate distributions (inverse transform, accept-reject, Metropolis-Hastings etc.) and it seems that we can sample from literally any valid distribution - is that true? Could you provide…
Tim
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