# Questions tagged [probability-theory]

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-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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### Is $P (C\mid A\;\text{and}\; B)$ the same as $\frac {P(A\cap B\cap C)}{P(A\cap B)}$?

Excuse me if this question is very basic: Is $P (C\mid A\;\text{and}\; B)$ the same as $\frac {P(A\cap B\cap C)}{P(A\cap B)}$?
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### Discrepancy of answers between differing computations of $E[e^{W_s}e^{W_t}]$ ($W_t$ being the Weiner process)

I was looking at another thread, and the following two distinct solutions to $E[e^{W_s}e^{W_t}]$ (assume that $W_0 = 0$ and $t>s$) were given, with both gving identical answers (I have slightly re-...
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### Understanding the “Découpage de Lévy” (or Levy decomposition)

I am looking for someone to help me understand what exactly the "Découpage de Lévy" (or Levy decomposition) of random variables is and why it works. The book I am reading is "The ...
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### plz help.. Sally is traveling from city A to city C by way of city B… [closed]

Sally is traveling from city A to city C by way of city B. There are 4 flights from A to B and 6 flights from B to C. How many different routes are possible Answer Choices A)10 B)12 C)24 D)15 Please ...
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### Density of noncentral $F$ distribution

Suppose $Z\sim N(\mu,1)$ and $V$ is independent of $Z$ with distribution $\chi^2_m$. Then $T=\frac{Z}{(V/m)^{1/2}}$ is said to have a noncentral $t$ distribution with noncentrality $\mu$ and $m$ ...
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### Guessing a random variable from moments

Define $\mu_m$ as follows: \begin{align*} \mu_{2m + 1} &= 0 \\ \mu_{2m} &= \frac{c^{2m}}{2m + 1}. \end{align*} Is there a continuous random variable $X$ such that $E[X^m] = \mu_m$? The ...
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### Why do homogeneous Markov chains with a transition probability matrix coinciding with a given stochastic matrix exist?

Why do homogeneous Markov chains with a transition probability matrix coinciding with a given stochastic matrix exist? And why would such a Markov chain, given a stationary distribution as the initial ...
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### Inferring bounds from joint typicality on three variables

Consider the following exercise from Cover and Thomas: And the given solution from the solutions manual: It is reasonably clear that these bounds are valid (one simply follows the counting argument ...
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### Under what conditions does the median of an $n$-sample of a random variable remain below its expectation

Let $X$ be a real-valued random variable satisfying $\mathsf E X = 0$ and $\mathsf P(X\le 0)\ge\frac12$. In other words, $X$ is a (centralised) random variable whose median is below its mean. For ...
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### If the median of a random variable is below its expectation, is this still true for a repeated sampling of this random variable?

Let $X$ be a real-valued random variable satisfying $\mathsf E X = 0$ and $\mathsf P(X\le 0)\ge\frac12$. In other words, $X$ is a (centralised) random variable whose median is below its mean. For ...
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### Effect of linear transformation on Gaussian Complexity

I'm reading Vershynin's tutorial on High dimensional Probability (linked at the end). Vershynin also has a book, where this and related concepts like Gaussian width are also discussed. In it, he ...
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### Avoidance of double loop of Monte Carlo integrations for nested expected values with indicator function [closed]

I have (potentially high dimensional) random variables $X$ and $Z$ and a constant $\rho \in [0,1]$. I can sample from $p(X,Z)$, $p(X|Z)$, $p(Z|X)$ , $p(X)$ and $p(Z)$, but don't know more about these ...
### Visual Representation of $X_n \sim U[0, \sin^{2}n]$
Trivially, I understand how the cdf/pdf of an uniform distribution with interval $[a,b]$ looks. But when there's a variable within my interval itself, say a sequence of RV's $X_n \sim U[0, \sin^{2}n]$...