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.
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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 ...
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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 $\...
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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) = ...
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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, ...
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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]...
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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'...
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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
$$
\mathbb{E}(\max\{0,X\})^...
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Proof that a moment generating function is unique
I am trying to understand the proof behind why the Moment Generating Function (https://en.wikipedia.org/wiki/Moment-generating_function) of a random variable is is unique.
For example, consider the ...
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Moment Generating Function of a Product of Random Variables
Suppose $X,Y$ are independent random variables, with respective moment generating functions (MGFs) $M_X(t)$ and $M_Y(t)$. It is known that the MGF of $XY$ is given by
$$\int_{-\infty}^{\infty} \int_{-\...
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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 ...
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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
$$M_{Y_1}(s) = E[\exp(...
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Identifying a moment generating function.
I am working on the following problem, and I am having trouble understanding why (a) is not a moment-generating other than it doesn't satisfy the general form of the MGF (i.e. $E[e^{tX}]$), and that (...
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moment generating function of the Borel distribution [closed]
Calculate the moment generating function of the Borel distribution given by $P(X=x; \mu) = \frac{e^{-\mu x} (\mu x)^{x-1}}{x!}$
with $\mu \in (0,1)$ and $x = 1, 2, 3,\ldots$.
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Compute $\mathbb{E}\left[\tau^2\right]$ when $\tau=\inf\{t>0|W_t-W_0\geq +a\textrm{ or }W_t-W_0\leq -b\}$
Let $W$ be a Brownian motion, $a>0$ and $b>0$ real constants. Let $\tau$ be the stopping time
$$
\tau=\inf\left\{t\geq 0:W_t-W_0\geq +a\textrm{ or }W_t-W_0\leq -b\right\}
$$
I need to compute $\...
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What is the expectation of this power series of random variables?
Let $(a_k)_{k\in\mathbb{N}}$ be iid random variables distributed uniformly in $(-1, 1)$, and consider, for some fixed $r \in [0, 1)$, the limiting random variable
$$ X = \lim_{n \to \infty} \sum_{k=0}^...
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