# Tagged Questions

Questions about maps from a probability space to a measure space which are measurable.

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### Exsistence of random variable $U[-1,1]$

I have problems with the following question: Let X be the random variable such that $X\sim U[-1,1]$. Does there exsist a random variable Y, independent to X, such that $X+Y\sim 2Y$? I thought that ...
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### Calculation of Conditional Expectation $\Bbb E[f(X)\mid Y]$

$\newcommand{\Cov}{\operatorname{Cov}}$and thank you for taking the time to read this. :) My question is about evaluating $\Bbb E[f(X) \mid Y]$ (a random variable in $Y$). There's plenty online (and ...
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### Skorohod representation theorem

Assume $X_n$ are random variables such that $\mathbb{P}(X_n \leq B)=1$ for some random variable $B$. Assume also $X_n \Rightarrow X$ ($X_n$ converges weakly to $X$). By the skorohod representation ...
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### Identifying the joint distribution from some values of $t \cdot X$

Suppose that $S$ is a subset of $\mathbb{R}^n$ and $X, Y$ are $\mathbb{R}^n$ valued RVs. We already know that $X$ and $Y$ are equidistributed iff $t \cdot X=^d t\cdot Y$ for all $t \in \mathbb{R}^n$. ...
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### Is Hoeffding's bound tight in any way?

The inequality: $$\Pr(\overline X - \mathrm{E}[\overline X] \geq t) \leq \exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right)$$ Is this bound (or any other form of hoeffding) tight in ...
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### A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}$ are i.i.d. $\sim$Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1$$ Then it follows ...
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### Relation between factor graph and conditional probability distribution

First, I'm from computer science. I don't know how to say this problem in a mathematical way. So please bear with me. The question Let say I have a factor graph illustrated in the figure. The ...
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### Characteristic functions of random variables (Poisson, Gamma, etc.)

My self-study in measure and probability theory as finally brought me to the subject of characteristic functions, and I have not handled these in the past with any rigor at all, so all of this is ...
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### Random variable for storing cost to get the target.

There is a simple game for a single player. Player's initial level is $n$ and player want to get level $m$. If player's level became the target level $m$, the game terminates. Player should pay $c_i$...
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### $P(|X_1+X_2|<x)\le P(|X_1|<x)$ for every independent centered continuous $X_1$ and $X_2$?

Let $X_1$ and $X_2$ be zero mean independent continuous random variables. Then, is it true that $P(|X_1+X_2|<x)\le P(|X_1|<x)$. The intuition is that summing independent variables increase ...
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### Trace of power of random matrix / sum of random variables with semicircle distribution

I want to calculate the expectation value for the trace of the $m$-th power of the $n\times n$ adjacency matrix $A$ of a large Erdos-Renyi random graph (without self-coupling, i.e., all diagonal ...
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We say that a random variable $X$ satisfies the $(\alpha,\beta)-$condition for some $\alpha>0$ and $\beta>0$ if $$\mathbb{P}(|X|<t)<\alpha t\text{ and }\mathbb{P}(|X|>t)<e^{-\beta t}\... 0answers 71 views ### PDF of Random Variable \sin\alpha \cdot \cos\beta with \alpha,\beta uniform As part of a bigger problem, I want to compute the probability density f_Z(z) of$$Z = \sin\alpha \cdot \cos\beta$$where \alpha, \beta are random variables, independently and uniformly ... 0answers 102 views ### Book on Convergence Concepts in Probability without Measure Theory I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ... 0answers 139 views ### Estimating the support of a probability density function The inverse moment problem deals with the reconstruction of a probability density function (PDF) of a random variable (RV) by means of its statistical moments. In the special case of the Hausdorff ... 0answers 49 views ### Rao-Cramer lower bound regularity condition and dominated convergence Let (\mathcal{X}, \mathcal{F}, (\mathbb{P}_\vartheta)_{\vartheta \in \Theta}) be a statistical model dominated by a sigma-finite measure \mu with Likelihood-function L(\vartheta, x) which is ... 0answers 221 views ### Central Limit Theorem for independent but non identically distributed random variables My question is the following: Given the sum of R.V.s, Z_N = X_1 + X_2 + ... +X_N , where X_i are independent, Rice distributed (X_i\sim Rice(\mu_i,\sigma) ), is there any way to approximate ... 0answers 75 views ### Simple random walk conditioning on non-return Consider a simple symmetric random walk on \mathbb{Z}, (S_t)_{t \geq 0}, with S_0=0. Let k and j be two positive integers. Let P_{k,j} be the probability that the walker hits the vertex k... 0answers 78 views ### Almost sure limit of \log(X_1 + X_2 + … + X_n) - \log(n) Let X_n be an i.i.d. sequence of positive random variables with expectation 2 and variance 1. What is the almost sure limit of$$\log(X_1 + X_2 + ... + X_n) - \log(n) as $n \to \infty$ Would it ...
Having a problem with the expectation of the maximum among $n$ independent random variables $X_1, X_2 \dots X_n$ all ~ the same class of distributions but not necessarily the same mean and other ...