# Tagged Questions

Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-...

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### For $X,Y$ random variables, $h$ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y)$ almost surely

Question in the title: For $X,Y$ random variables, $h$ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y)$ almost surely My main problem is that I don't even understand what $E (Xh(Y)|Y)$ means.....
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### Is it true that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$?

I was wondering if we can show that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$ in general? Here $X_1$ and $X_2$ are independent but may not follow the same distribution. Any hint is much ...
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### What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
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### $E(X_1|X_1+X_2=k)$ increases with $k$?

$X_1$ and $X_2$ are independent, but they may not follow the same distribution. I want to know whether $E(X_1|X_1+X_2=k)$ increases with $k$. I guess this is correct, but is there a proof or counter ...
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### Are there any modern mathematicians whose research interest is in “Probability Theory”? [closed]

I have seen professors in universities list "stochastic calculus", "stochastic analysis", "stochastic processes", "stochastic geometry" and "applied probability" as research interests, but are there ...
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### Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
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### distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
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### Gram matrix for a random variable vector space with inner product?

I am wondering if it is possible to construct a list of binary valued random variables, $\{\bf{X}_1,\bf{X}_2,\bf{X}_3\}$ and define a Gram-like matrix like \begin{bmatrix} \langle\bf{X}_1,\bf{X}_1\...
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### Convolution of a function with itself n times convergence to bell curve [duplicate]

If we have a piecewise function defined as $f(x) = \begin{cases} 1, & \text{0$\lex\le$1} \\ 0, & \text{otherwise} \end{cases}$ Explain how the convolution of $f$ with itself for $n$ ...
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### Moment generating function of $X+Y$ using convolution of $X$ and $Y$

Given that the pdf of $X+Y$ is the convolution of pdfs $X$ and $Y$; show that $M_{X+Y}$ is $M_XM_Y$ where $M$ is the moment generating function. $X and Y$ are independent and continuous. I am confused ...
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### Convergence of expectations of a sequence of exponential random variables.

Suppose $\{X_n\}$ is a sequence of exponentially distributed random variables, where $X_n$ has mean $1/\lambda_n$. Suppose that $\lim_{n\to\infty}\lambda_n = \lambda>0$. Let $X$ be exponentially ...
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### Sequence of Erdos-Renyi random graphs convergent with probability 1

Definitions Let $H,G$ be finite simple graphs. Then the density of $H$ in $G$, denoted $d(H,G)$, is defined as the probability that a randomly chosen $|H|$-tuple of vertices of $G$ induce a graph ...
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### Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
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### CGF determines the distribution

It is well known, that if the domain of the mgf of a random variable $X$ contains an interval around zero, then the distribution is completely determined by its moments. However consider a Levy-...
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### Probability question, can I reset the window or not

There is a wall street banker. The banker invests in a kind of share called as options. The main features of this share is as follows: You make a bet with a specified amount of information as to ...
I'm reading a paper from Creal, Koopman, Lucas "Univariate Generalized Autoregressive Score Volatility Models" and I'm stuck with this computation.  -\operatorname{E}_{t-1} \left[ \frac{\partial^...