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### Probability of two vertices being connected by some path in a random directed graph

Consider a BFS exploration process, which proceeds in $k$ stages. Put $V_0 = \{u\}$. Given $V_0,\ldots,V_i$, explore all edges from $V_i$ to $V \setminus \bigcup_{j=0}^i V_j$ (where $V$ is the set of ...

### Upper bound to number of closed itemsets

The bound is $2^{\min(n, m)}$. It is an upper bound because no two "formal concepts" (i.e., closed itemsets with their respective transaction sets) can have the same subset of items or the same subset ...

### computing maximal bit density over a FSM

The supremum bit density will either be achieved by a finite word $v$ in the language, or by the limiting bit density of some sequence $u v w, u v^2 w, u v^3 w, \ldots$ of words in the language, which ...
### Probability of random variable $X$ less than $max(Y_i)$
We have $\Pr[\max(Y_1,Y_2) \leq k] = \Pr[Pois(\lambda) \leq k]^2 = e^{-2\lambda}(\sum_{j=0}^k\frac{\lambda^j}{j!})^2$. Therefore, \$\Pr[X \geq \max(Y_1,Y_2)] = \sum_{k \in \mathbb N} \Pr[X = k] \cdot \...