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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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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 ...
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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 $\gamma>0$...
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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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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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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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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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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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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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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 $$ \mathbb{E}(\max\{0,X\})^...
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I am trying to understand the proof behind why the Moment Generating Function (https://en.wikipedia.org/wiki/Moment-generating_function) of a random variable is is unique. For example, consider the ...
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Suppose $X,Y$ are independent random variables, with respective moment generating functions (MGFs) $M_X(t)$ and $M_Y(t)$. It is known that the MGF of $XY$ is given by $$\int_{-\infty}^{\infty} \int_{-\...
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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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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 $$M_{Y_1}(s) = E[\exp(...
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I am working on the following problem, and I am having trouble understanding why (a) is not a moment-generating other than it doesn't satisfy the general form of the MGF (i.e. $E[e^{tX}]$), and that (...
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Calculate the moment generating function of the Borel distribution given by $P(X=x; \mu) = \frac{e^{-\mu x} (\mu x)^{x-1}}{x!}$ with $\mu \in (0,1)$ and $x = 1, 2, 3,\ldots$.
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Let $W$ be a Brownian motion, $a>0$ and $b>0$ real constants. Let $\tau$ be the stopping time $$ \tau=\inf\left\{t\geq 0:W_t-W_0\geq +a\textrm{ or }W_t-W_0\leq -b\right\} $$ I need to compute $\...
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Let $(a_k)_{k\in\mathbb{N}}$ be iid random variables distributed uniformly in $(-1, 1)$, and consider, for some fixed $r \in [0, 1)$, the limiting random variable $$ X = \lim_{n \to \infty} \sum_{k=0}^...
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Let $$X_t = X_0 + \mu t + \sigma W_t$$ where $W_t$ is a standard brownian motion, $\mu \in \mathbb{R}$ and $\sigma > 0 $. Let $\tau$ denote the first time of hitting an upward barrier, e.g.: $$ \...
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Definition Let X be a real value random variable. The moment-generating-function is defined by $M(t):=\int_{\mathbb{R}}e^{tx} dx$. Exercise: Suppose X,Y are independent random variables. Show that $...
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Let $X$ be a continuous real random variable. For $t \in \mathbb{R}$ and $i^2=-1$, $\mathbb{E}[e^{itX}]$ is the characteristic function of $X$. For $s \in \mathbb{R}$, $\mathbb{E}[e^{sX}]$ is the ...
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Let $(X_n)$ be a sequence of random variables and let $X$ be another random variable. Assume that the MGFs of $X_n$ converges pointwise to the MGF of $X$ at positive points. That is, for all $t > 0$...
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Let $S_n=\xi_1+\dots+\xi_n$ be a random walk. Suppose $\psi(\theta_0)=E[e^{\theta_0\xi_i}]=1$ for some negative $\theta_0$ and $\xi_i$ is not a constant. In this case, $X_n=\exp(\theta_0 S_n)$ is an ...
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From Blitzstein and Hwang (2nd edition): Let X be the total from rolling 6 fair dice, and let X1, . . . , X6 be the individual rolls. What is P (X = 18)? The PGF of X lets us count the cases in a ...
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If one plots a bootstrap distribution of "resample means" obtained by resampling a single, real sample from a population distribution, the mean of that plot gives no more information than ...
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Suppose we have two bounded non-negative random-variables $X,Y$. Suppose that $\mathbb{E}[X]=\mathbb{E}[Y]=1$ and their other moments obey $\mathbb{E}[X^m] \geq \mathbb{E}[Y^m]$ for all integer $m \...
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I am reading in wiki that for a non-negative random variable X we get $$\operatorname P \left(X > a\right) \geq \sup_{t > 0 \text{ and } M(t) \geq e^{ta}} \left( 1 - \frac{e^{ta}}{M(t)} \right)^...
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For a Rademacher r.v. X that is defined as $$Pr[X = k]=\begin{cases}\frac{1}{2}, \text{ for k=1} \\ \frac{1}{2}, \text{ for k=-1} \\ 0, \text{ otherwise}\end{cases} \tag{1}$$ or alternatively as $$Pr[...
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Given that the moment generating function of the random variable X is $M_X(t)=\frac{1}{1-2t}$, calculate the expectation of $Y^n,n\in \mathbb{N} $, where $Y=365 \times 0.3^X$. Here is my attempt. Use ...
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Let $X_1, \ldots, X_n$ random variables i.i.d. with geometric distribution ($k \geq 1$) with parameter p. Define $Y_p := X_1 + \ldots + X_n$. Show that $p Y_p$ converges in distribution distribution ...
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From Taylor series, if I need to get the k-th moment, I need to find the k-th derivative of the Moment Generating function. If I have $X \sim \text{Uniform}(0,1)$, the MGF $M_{X}(s)$ is; $$M_{X}(s) = \...
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While doing a study on the growth metrics RRRGR (Reverse of Relative of Relative Growth Curve), I'm stuck on the exact distribution of the growth metrics $ W_{ji} = ln \left( \dfrac{R_{ji} (t)}{R_{ji} ...
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Let $S$ be a Poisson Binomial random variable, i.e. the sum of $n$ independent Bernoulli random variables with parameters $p_1, ..., p_n$. I'd like to know the expected value of $\frac{1}{1+S}$, ...
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At first, this looked like an easy task, starting with usual expression for variance, such as: $\operatorname{Var}(S^2)= E(S^4) - (E(S^2))^2$. Once again, herein I want to define $S^2=\frac{\sum_1^n (...
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$X$ and $Y$ are unbounded random variables, if $X^n$ and $Y^n$ are integrable for all n, and $E(X^n) = E(Y^n)$ for all $n\ge0$ must $X$ and $Y$ have the same distribution? My intuition says no. I’ve ...
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Is the method of moments an if an only if statement when the limiting distribution is the standard normal and the moments of the sequence are all finite? That is, suppose we have a sequence $X_n$ ...
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The generalized moment generating function (MGF) is defined (at least in some of the signal processing literature) by \begin{equation} M_X^{(r)}(t) \doteq {\mathbb E}(X^r\exp{(tX)}. \end{equation} The ...
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I am trying to prove the following inequality for an arbitrary random variable $X$ and a constant $c$: $$P(X\ge c)\le \min_{s\ge 0}e^{-sc}M_X(s),$$ where $M_X(s)$ is the moment generating function of $...
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Let $(X, Y)$ be an absolutely continuous random vector and $M_{X, Y}$ be the bivariate moment generating function of $(X, Y)$. I want to show that $X$ and $Y$ are independent if and only if $$M_{X, Y}\...
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If $X$ is sub-Gaussian random variable with variance proxy $\sigma^2$, i.e., $E(X) = 0$ and $E\{ \exp(s X) \} \leq \exp( \frac{\sigma^2 s^2}{2} )$ for $\forall s \in \mathbb{R}$, then how to show that ...
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I am searching the definition of the $5^{th}$ unbiased cumulant estimate. Let $K_j$, be the $j$-th unbiased cumulant estimate of a probability distribution, based on the sample moments. Let $m_j$ be ...
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Let $X$ be a non-negative random variable bounded on $[0,1]$. Is it true that $\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$ for some constant $k$? If not, are there any minimal assumptions on $X$ where this ...
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I recently had an assignment where I had to prove that, if $X,Y,Z$ are all independent random variables and $X+Z$ has the same distribution as $Y+Z$ then $X$ has the same distribution as $Y$. These ...
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For a random variable $X$, the cumulant generating function $CGF_X$ is defined as $CGF_X(t)=\log Ee^{tX}$, and the nth cumulant $k_n(X)$ is defined as the coefficient of $t^n/n!$ in the corresponding ...
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I tried using the moment generating function. I got: $M_{X}(t)=E(e^{tX})=\Sigma_{n=0}^{\infty} (n+1)(3t)^{n}$. But I don't understand which distribution would it be.
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Showing that $(e^{\lambda B_t - \frac{1}{2}\lambda^2t})_{t\geq 0}$ is a $\mathbb{R}$-valued martingale Let $B$ be a standard $\mathbb{R}$-valued Brownian motion and let $\lambda\in\mathbb{R}$. From $...
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Let $X \in \mathbb{R}^d$ be a zero-mean multivariate Gaussian, with independent components, i.e., $X \sim \mathcal{N}(0,\Sigma)$, with $\Sigma = diag(\sigma_1^2,\ldots,\sigma_d^2)$ a diagonal matrix. ...
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Let $X_n$ be the size of the $n$-th generation of a branching process, with family size probability generating function $G(s)$. Let $X_0 = 1$. Suppose the family-mass size mass function is $P(X_1 = k) ...
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Let $X$ be a non-negative r.v., prove that \begin{equation} \inf_{k\in\mathbb{Z}_+} \frac{\mathbb{E}[X^k]}{t^k}\leqslant \inf_{\lambda\geqslant 0}\frac{\mathbb{E}[e^{\lambda X}]}{e^{\lambda t}},\;\...
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Question Define $\mathbf{a} = (a_1, \ldots, a_p)$ where $p$ is a positive integer and the $a_l$ are i.i.d $\text{Uniform}(-1,1)$ random variables. Fix a unit vector $w \in \mathbb{R}^p$. Consider the ...
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Suppose I have $N$ iid random vectors $\sigma_1,\ldots,\sigma_N$ that are uniformly distributed in $S^1$. Let $\bar{\sigma}_N:=N^{-1}(\sigma_1+\cdots+\sigma_N)$ denote the sample average. I am ...
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