Questions tagged [moment-generating-functions]
For questions relating to moment-generating-functions (m.g.f.), which are a way to find moments like the mean$~(μ)~$ and the variance$~(σ^2)~$. Finding an m.g.f. for a discrete random variable involves summation; for continuous random variables, calculus is used.
1,358 questions
5
votes
1
answer
290
views
Question regarding Jensen's inequality when it comes to logarithm
Let $X$ be a real-valued random variable, and define its moment generating function (MGF) as
$$
M_X(s) = \mathbb{E}[e^{sX}],
$$
where $\mathbb{E}[\cdot]$ denotes the expected value of the random ...
- 191
0
votes
0
answers
35
views
Concentration for Markov chain with spectral gap
Sub-Gaussian concentration for reversible Markov chains with spectral gap
Setup.
Let $(X_i)_{i\ge1}$ be a stationary, $\pi$-reversible Markov chain on a measurable space with spectral gap $\gamma>0$...
- 178
1
vote
1
answer
103
views
An upper bound for the moment generating function $f(t) = \mathbb{E}[e^{tX}]$ when the mean, variance and upper bound of $X$ are given
Suppose a real random variable $X$ has an upper bound $c > 0$ but has no lower bound. $\mathbb{E}[X] = 0$ and $\mathbb{E}[X^2] = \sigma^2$ ($\mathbb{E}[]$ denotes the expectation of random ...
- 43
4
votes
2
answers
118
views
A random variable $X$ is sub-Gaussian iff $\exists C>0: \mathbb{E}[e^{Xt}]\le C\exp(Ct^2)$
This is exercise 1.1.4 from Tao's "Topics in Random Matrix Theory". It asks the reader to prove that a real-valued random variable $X$ is sub-Gaussian iff there exists such $C>0$ that $\...
- 1,506
0
votes
1
answer
146
views
Can two non independent random variables be split into independent ones?
Let $(X,Y)$ be a random vector with joint moment generating function
$$M(t_1,t_2) = \frac{1}{(1-(t_1+t_2))(1-t_2)}$$ Let $Z=X+Y$. Then,
Var(Z) is equal to: (IIT JAM MS 2021, Q21)
Using $M_{X+Y}(t) = ...
- 2,674
1
vote
0
answers
44
views
Moments of a sum of iid random variables with mean 0 and variance 1
The method of moments Wikipedia page says that odd moments of the sum $S_n$ of iid random variables $X_i$ (each mean 0 and variance 1) vanish. While even moments are all finite and have closed form, ...
- 522
2
votes
0
answers
59
views
What is the most efficient algorithm for computing $E[x_1 x_2 \cdots x_n]$ in a multivariate normal distribution? [closed]
I am working with a multivariate normal distribution $\mathbf{x} = [x_1, x_2, \ldots, x_n] \sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, and I need to compute the expectation $E[x_1 x_2 \cdots x_n]...
- 21
2
votes
1
answer
251
views
What does Chernoff bound want to point out? Intuition
e.g. Chebyshev's bound $P(|x-\mu| \geq a) \leqslant \frac{\sigma^2}{a^2}$ tells us the upper-bound probability that $X$ deviates from its mean (in absolute value) by more than a certain amount.
I can'...
user1502876
0
votes
0
answers
48
views
On the finiteness of moments
Consider a random variable $X$ with full support and assume
$$
\mathbb{E}(\max\{0,\exp(\beta X)\})^p<+\infty
$$
for some $\beta>0$ and $p>0$.
Does this imply that
$$
\mathbb{E}(\max\{0,X\})^...
- 414
3
votes
1
answer
256
views
Proof that a moment generating function is unique
I am trying to understand the proof behind why the Moment Generating Function (https://en.wikipedia.org/wiki/Moment-generating_function) of a random variable is is unique.
For example, consider the ...
- 4,183
1
vote
1
answer
70
views
Moment Generating Function of a Product of Random Variables
Suppose $X,Y$ are independent random variables, with respective moment generating functions (MGFs) $M_X(t)$ and $M_Y(t)$. It is known that the MGF of $XY$ is given by
$$\int_{-\infty}^{\infty} \int_{-\...
- 2,237
4
votes
0
answers
83
views
What "methods" does a multivariate Moment-Generating Function provide?
I don't think the Probability and Random Variables course I studied really touched on multivariate Moment-Generating Functions at all, and Wikipedia appears surprisingly silent on this question. The ...
- 2,184
1
vote
0
answers
61
views
Proof of central limit theorem by moment generating functions and the assumption on $X_i$
I was reading the the proof of CLT by MGF in this answer
Let $Y_i$'s be i.i.d random variables with mean 0 and variance 1, and in the original answer, by Taylor expansion we have
$$M_{Y_1}(s) = E[\exp(...
- 925
2
votes
1
answer
57
views
Identifying a moment generating function.
I am working on the following problem, and I am having trouble understanding why (a) is not a moment-generating other than it doesn't satisfy the general form of the MGF (i.e. $E[e^{tX}]$), and that (...
- 169
-1
votes
1
answer
122
views
moment generating function of the Borel distribution [closed]
Calculate the moment generating function of the Borel distribution given by $P(X=x; \mu) = \frac{e^{-\mu x} (\mu x)^{x-1}}{x!}$
with $\mu \in (0,1)$ and $x = 1, 2, 3,\ldots$.
