# Tagged Questions

Moment Generating Function

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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 ...
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### 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 ...
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### Steps to calculate the MGF of (1+sum of gamma distribution) over gamma distributions

What is the best way or steps to calculate the moment generating function (mgf) of (1+sum of gamma distribution) over gamma distributions? Is it the same way to derive beta distributions? I am trying ...
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### MGF technique for exponential / Chisquared

Let $X$, $U$, $V$, $W$, $Y$ be independent exponential random variables with respective means $\frac{1}{6}, \frac{1}{4}, \frac{1}{3}, 1, \frac{2}{3}$. Use the variables $X, U, V, W, Y$ to construct a ...
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### Probability density and moment generating [duplicate]

Please show some working when substituting the equation. Thanks
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### Properties of moment-generating functions

I am new to statistics and I happen to came across this property of MGF: Let $X$ and $Y$ be independent random variables. Let $Z$ be equal to $X$, with probability $p$, and equal to $Y$, with ...
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### The mgf and cf of Student's t distribution

A student's t distributed rv $X$ has characteristic function but no moment generating function. I wonder if cf(X)=$E[e^{itX}]$, why we cannot take $t=-iu$ to get the mgf $E[e^{uX}]$? (This question ...
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### How to derive mean of chi square variable function using the MGF?

I am working through past examination questions from the Royal Statistical Society and came across this one from 2009 in Module 5 (Question 2(i) and Solution): The random variable $X$ has a ...
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### Welch Test Statistic

I've read journal article On the Comparison of Several Mean Values: An Alternative Approach (Welch, 1951). How can I derive the MGF of the Welch statistic? Are there other methods to get the Welch ...
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### 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 ...
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### What is the expectation of exponential of the product of two random variables?

I am looking for examples of probability distributions that would allow me to characterize the distribution (at least approximately) and to compute the first two moments exactly of: $$e^{aXY}$$ ...
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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?
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### Is the Poisson distribution stable and are there inversion formulas for the MGF?

First, I have a question about whether the Poisson distribution is "stable" or not. Very naively (and I'm not too sure about "stable" distributions), I worked out the distribution of a linear ...
I'd be grateful for any hints or help with this question: Let $X$ follow the Weibull distribution with pdf $f(x)=\beta x^{\beta-1}e^{-x^{\beta}}$ on $x>0$ with $\beta>0$. Show that ...
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 ...