All Questions

Suppose we have a probability distribution $p(\cdot)$ supported on the integers between $-m$ and $m$ for some positive integer $m$, with $\sum_k kp(k) = 0$. Suppose furthermore that all $p(k)$ are ...

I am looking for an explicit formula for a sequence. The sequence is generated as follows: There is a tournament with $10$ teams. In the beginning, all teams have a 0-0 win-loss record. The teams are ...

How to prove the following identity $$ \sum_{n=1}^{\infty} \frac{n^{n-1} e^{-n}}{n!} = 1$$ analytically (which can be confirmed with $Mathematica$)? The standard trick for geometrical series does not ...

Fix $n, k$. Let $$ C^{n,k}_r =\frac{1}{r!} \binom{n}{\underbrace{k, \ldots, k}_{\text{r times}}, n-rk} = \frac{n!}{r!(k!)^r(n - kr)!} $$ be the number of ways to form $r$ disjoint subsets each of ...

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...

How can one solve the following recurrence relation? $f(n) = 1 + \frac{1}{2^n} \sum_{k = 0}^n {{n}\choose{k}} f(k)$ $f(0) = 0$ As it happens, I can show $f(n) = \Theta(\log n)$ through other means (...