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I have a Bernoulli trial with success rate $p$ and failure rate $1-p$ the odds of $k$ successes is $\binom{N}{k} p^k (1-p)^{N-k}$. I need to evaluate an integral $$ \int_0^1 dp p^k (1-p)^{N-k} = \f …

I found in some lecture notes that Brownian motion is defined by its Fourier series: $$ B(t) = \sum_{m \in \mathbb{Z},\; m \neq 0} \frac{a_m}{m} e^{2\pi i \,mt} $$ Then I would get that its derivati …

The Krawtchouk ensemble is defined by a weight: $w(x) = \binom{K}{x}p^x q^{K-x} $ and in fact it comes from a conditioned random walk on $\mathbb{Z}^N$. It is a probability measure on the set $\{ 0, …

Are there many implementations of the "domino shuffling" algorithm as found in William Jockusch, James Propp amd Peter Shor's Random Domino Tilings and the Arctic Circle Theorem math.CO/9801068? This …

A determinantal process on the line is a random collection of points on $\mathbb{R}$ such that the probability of $x_1, \dots, x_n$ lying on the random set is $\det (K(x_i, x_j))_{(i,j)}$. Examples …

Some time ago I asked about the odds 2 group elements commute. I wonder about the odds that 3 group elements commute. Is there a "closed" formula for the sum $$ \frac{1}{|G|^3} \sum_{g,h,k} \delta( …

The Cohen-Lenstra statistics describe how often a prime divides the class number of quadratic number field $\mathbb{Q}[\sqrt{d}]$ $$ \mathbb{P}\big[h(d) \not\equiv 0\; (\mod p) \big] = \prod_{k \geq …

n-point correlations of Gaussian random variables can be simplified with Wick expansion. $$ \langle x_{i_1} x_{i_2} \dots x_{i_{2n-1}} x_{i_{2n}} \rangle = \int_{\mathbb{R}^n} x_{i_1} \dots x_{i_{2n} …

An alternating permutation of {1, ..., n} is one were π(1) > π(2) < π(3) > π(4) < ... For example: (24153) is an alternating permutation of length 5. If $E_n$ is the number of alternating permutations …

Motivated by unfoldings of the dodecahedron in How To Fold It -- How many (labeled or unlabeled) unfoldings of the 1 x 1 x n stack of unit cubes are there? JORourke (4Nov16): John's original image …

I am reading Zeros of Gaussian Analytic Functions by Mikhail Sodin and he gives an much-too-easy proof of density of zeros of a Gaussian Analytic function. Definition A Gaussian analytic function …

Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of a random Unitary matrix. I am interested in the expected value: $$\mathbb{E}_{X \in U(N)}\left[ \prod_{i=1}^n \frac{1}{(1 - \lambda_i)^2}\righ …

I'm reading about amenable groups. What are explicit examples of nonabelian discrete amenable groups other than finite groups? Perhaps a group presentation or matrix representation would be useful.

I was reading about the Hasse-Weil bound for the number of points in on a curve over the finite field $\mathbb{F}_q$. $$ \big| |C(\mathbb{F}_q)| - (q+1) \big| \leq 2g \sqrt{q} $$ However, this rem …

I am reading Andrew Granville's Anatomy of Integers and Permutations where it is argued the factorization of a permutation into disjoint cycles is analogous to the factorization of a number into prime …