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Results tagged with pr.probability
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user 4354
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
19
votes
Accepted
What kind of probability distribution maximizes the average distance between two points?
The uniform distribution on the circle is optimal.
Every probability measure on the disc can be approximated by the sum of atomic measures with equal wieghts, that is, by measures of the form $\frac1 …
- 32.4k
12
votes
Accepted
Product of random diagonals on the unit circle
To 2, the answer is yes. This is easy. Re-enumerate points according to their cyclic order. For every fixed $k$ ($1\le k<n$), the product
$$
D_k:= \prod_i |P_i P_{i+k}|
$$
(where the indices are take …
- 32.4k
3
votes
Accepted
Distances between and among points in a region
One can show that $F(X)> c\cdot G(X)^{-2}$ for some $c>0$, provided that $G(X)$ is sufficiently small.
Let $G(X)=\varepsilon$. Divide the square into $\approx(100\varepsilon)^{-2}$ small squares of s …
- 32.4k
6
votes
Accepted
Scale random variables in a way they have equal probabilities of being minimal
Such simple adjustment is not possible. First, take the logarithm: consider new random variables $y_i=\log x_i$. Their ordering is the same, but correction is now additive rather than multiplicative: …
- 32.4k