Moment Generating Function
0
votes
1answer
35 views
Probability density and moment generating [duplicate]
Please show some working when substituting the equation. Thanks
3
votes
2answers
112 views
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 ...
0
votes
1answer
42 views
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 ...
3
votes
1answer
97 views
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 ...
2
votes
0answers
59 views
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 ...
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 ...
4
votes
1answer
803 views
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:
$$
e^{aXY}
$$
...
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
2answers
378 views
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 ...
3
votes
1answer
761 views
Weibull moment generating function and Gamma function
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
...
8
votes
1answer
371 views
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 ...
