# Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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### Minimal example of Simpson's paradox

Let's say that a finite probability space $(\Omega,\mathscr P(\Omega),P)$ has Simpson's property if you can find events $A,B,C\in\mathscr P(\Omega)$ such that $P(C) \in (0,1)$. $A$ and $B$ are ...
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### Difference between Bernoulli random variables

Given are $n$ independent Bernoulli random variables with parameters $p_1,\dots,p_n$. We want to split them into two parts so as to minimize the expectation $\mathbb{E}[|X-Y|]$, where $X$ is the sum ...
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### Optimal Transport: Showing Map is Optimal (simple question: more of an analysis/measure theory question)

I'm reading these lecture notes (not for homework, for fun) http://www.maths.gla.ac.uk/~gbellamy/LMS/BourneLectures.pdf I'm trying to figure out what it means for a transport means to be optimal. I'm ...
Let $(X,Y)$ be a point chosen at random from the triangle $\{x,y:0\leq x\leq y\leq 1\}$. $f_{X,Y}(x,y)=2$ if $(x,y)$ is in the triangle, and it is 0 otherwise. Find the probability density function ...
Prove that if state $i$ in a class has period $p$ then all states in that class have period $p$. The proof is given on this answer is this: One way to define the period of state $i$ is as the ...