Let $I$ be a nonempty directed set, and $H_i \le G_i$ both be directed sequence of groups living in some ambient group. Is there a nice formula for $|\bigcup_{i \in I} G_i : \bigcup_{i \in I} H_i|$, perhaps involving a limit or supremum of the $|G_i : H_i|$? I must confess, I don't really have mu...

Update: The tag limits-colimits was created by Martin. So it seems that the best course of action is now to merge these two tags, or add them as synonyms. Do we need the direct-limit and inverse-limits? I have no opinion on the matter, but maybe those should be merged into a single tag someho...

Suppose we have $p$ random variates :- $X_1, \cdots , X_p$. $n$ observations involving each random variable are made . Thus, the $jth $ trial or $jth$ observation is a vector of the form : $[x_{j1},x_{j2},\cdots, x_{jp}]^T$ where $x_{ji}$ means the value in the $jth$ trial for the $ith $ random v...