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Mr John Pond, Esq. in the Philosophical transactions of the Royal Society of London, 1806, part II, p431: "this quantity was very uniformly distributed though the intermediate arc". Rev. Baden Powell …

I just thought I'd record a cute "answer". $$ \operatorname{Prob}(X-Y=k) = \frac{1}{2\pi} \int_{-\pi}^\pi \cos(k\theta)\,(2P\cos\theta+1-2P)^n\,d\theta, $$ where $P=p(1-p)$. If $k$ is not …

Interestingly, the probability that $t$ occurs for a standard die $n=6$ as $t\to\infty$ is $\frac27$, with exponential convergence towards that value. This is a standard generating function problem. T …

Restricting $Y$ to an annulus doesn't seem useful as any bounds are likely to be satisfied also inside the annulus. A bound with $Y$ restricted to a disk, or more generally to a region with bounded di …

(Partial answer) In GF(2), row and column permutations preserve the determinant. The equivalence classes under those operations are non-isomorphic bipartite graphs with $n$ vertices on each side and $ …

My paper here (Adv. Appl. Prob., 21 (1989) 475-478), Theorem 2, provides an estimate over all values of the parameters with relative error that is $o(1)$ if either $\sigma\to\infty$ or $x\sigma\to\inf …

Consider the model where a random graph is made by adding one edge at a time chosen uniformly at random from edges not yet present. Łuczak, Pittel and Wierman (1994) showed that there is a function $f …

Take a bernoulli variable with weight $\log^{-4} n$ at $-\log n$ and weight $1-\log^{-4} n$ at $\log n/(\log^4 n-1)$. Then the mean is 0, the variance is $O(\log^{-2} n)$, the 4th moment converges to …

(Not a complete solution.) An interesting property is this: For an edge $uv$, the distribution of $b(u)+b(v)$ conditioned on $b(u)$ is the same as the unconditional distribution (namely uniform). From …

As Jukka writes, exact solutions are unattainable (or uselessly complicated) except for very low degrees and special cases. An example of such a special case is regular graphs of degree $d$, where the …

If a sequence of 1-dimensional log-concave integer-valued distributions satisfies a Central Limit Theorem (CLT) and has variance going to $\infty$, then it satisfies a Local Central Limit Theorem (LCL …

I'm not sure why you ask for "distinct integers" when sampling without replacement guarantees distinctness. Let $q_i=1-p_i$. The ordinary generating function $$F(u,y) = \prod_{i=1}^n (q_i+p_i uy) \pr …

Let $X_1,X_2,\ldots,X_n$ be iid random variables with hypergeometric distribution. To be specific, $$ \mathrm{Prob}(X_1=i) = \frac{\binom{N}{i}\binom{M-N}{m-i}}{\binom{M}{m}}.$$ Let $S=X_1+\cdots+X_n$ …

For fixed $k\ge 3$, $\frac{v(k,n)_{\ge R}}{v(k,n)}\to 1$. First note that most such graphs have trivial automorphism groups, so it doesn't make a difference whether you ask about isomorphism classes o …

Divide the vertex set into a fixed number of parts, in any way you like (such as randomly). Choose two probabilities $p_1,p_2$, where $p_1\le n^{-\varepsilon}$ for some fixed $\varepsilon\gt 0$ and $p …