Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
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Covariance of $Z'Vb$ given that the rows of V are i.i.d.

Suppose that we have the following entities $$ \underbrace{Z}_{n\times k},\quad\underbrace{V}_{n\times L},\quad \underbrace{b}_{L\times 1}. $$ $Z$ and $b$ are nonstochastic whereas we assume that the ...
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Proving two random variables differ with positive probability

EDIT: Despite the help of the posters below, I'm still confused. I'm rephrasing the question slightly. Can someone hep me with rephrased problem: Suppose that $X$ is a random vector and $Y$ a random ...