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Category Archives: Probability
Probability that a random inscribed triangle contains another circle
Suppose is the unit circle centered at on the plane and is a concentric circle with radius (), what is the probability that a random triangle inscribed in contains in its interior? This is a problem raised by one of … Continue reading
Posted in Calculus, Geometry, Probability
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Noncommutative probability II: independence
Recall that two random variables and are said to be independent if for all Borel sets . We don’t have a probability measure behind the abstract definition of noncommutative probability space, so we cannot define independence in this way. However, … Continue reading
Posted in Combinatorics, Probability
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Noncommutative probability I: motivations and examples
In many areas of mathematics, one can usually assign a commutative algebraic structure to the “space” that one is studying. For example, instead of studying an (affine) algebraic variety, one can study algebraic functions on that variety, which gives rise … Continue reading
Posted in Functional analysis, Probability
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Zeros of random polynomials
Given a polynomial , where the coefficients are random, what can we say about the distribution of the roots (on )? Of course, it would depend on what “random” means. Here, “random” means that the sequence is an i.i.d. sequence … Continue reading
Posted in Algebra, Complex analysis, Potential theory, Probability
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Why is a² + b² ≥ 2ab ?
This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again). How can one … Continue reading
Posted in Calculus, Discrete Mathematics, Geometry, Inequalities, Linear Algebra, Probability
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Two little remarks on sphere
1. “Take a sphere of radius in dimensions, large; then most points inside the sphere are in fact very close to the surface.” David Ruelle Let and . Let The fraction of the volume of to the volume of is … Continue reading
From combinatorics to entropy
Let and . Then by Stirling’s formula. This is probably well-known to people who have studied statistical physics, but some of us including myself may not be very aware of this kind of relation. I wonder if this was the … Continue reading
Martingale Theory III: Optional stopping theorem
This is a sequel to Martingale Theory II: Conditional Expectation. Our aim here is to prove the main theorems about discrete time martingales. More advanced probability texts (e.g. those on stochastic calculus) assume that these theorems are well known to … Continue reading
Posted in Probability
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Potential theory on finite sets
Motivation Consider the following Dirichlet problem on the unit square : on where is a given function on the boundary. To get an approximate solution, we may replace the unit square by a grid. Let be large and set . … Continue reading
Posted in Potential theory, Probability
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Estimating the probability of grad school admission
We start with a naive model of the situation. Model 1. Suppose that I apply for schools , , …, . I estimate that I will be admitted to school with probability . Thus, if denotes the event that I am admitted … Continue reading
Posted in Miscellaneous, Probability
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An indicator approach to discrete probability
The purpose of this elementary post is to illustrate that much of discrete probability can be analyzed in terms of indicator functions, linearity and independence. These ideas are well-known in probabilitly, but the following approach is seldom seen in elementary … Continue reading
Posted in Probability
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