I want to write the following sentence formally:
The sequence $S$ contains elements of the set $A$. The probability value $P(a)$ for an element $a$ is defined as the number of its occurrences in the sequence $S$, divided by the count of all its elements.
I can write it the following manner:
$$ S = (s_1, s_2, …, s_n) : s_i \in A.$$
$$ P(a) := {{ \left| \lbrace i \in \lbrace 1, 2, …, n \rbrace : s_{i} = a \rbrace \right| } \over {n}}, \text{ given } n > 0\text{ and }a \in A. $$
It's, however, quite long and rather not elegant. Is there a simpler way to write this?
Edit:
There's always a solution, which involves breaking the formula to smaller parts:
$$ \text{Let } C(x) = \left| \lbrace i \in \lbrace 1, 2, …, n \rbrace : s_i = x \rbrace \right|.$$
$$ P(a) := {C(a) \over n}. $$
It's more readable, but it's still not what I'm searching for…
Best Answer
If you are willing to use the "Iverson bracket notation", popularized by Knuth and others, you can say $$C(x) = \sum_{i=1}^n [s_i = x]$$
Here $[\ldots]$ are the Iverson brackets. $[P]$ is defined to be 1 if $P$ is true, and 0 if it is false.
People do sometimes use the Kronecker delta for this: $\delta_{ij}$ is defined to be 1 if $i=j$ and 0 if $i\ne j$, so you would have:
$$C(x) = \sum_{i=1}^n \delta_{xs_i}$$ or $$C(x) = \sum_{i=1}^n \delta(x, s_i)$$
but I think the Iverson bracket is more straightforward.
Most straightforward would be to write
The idea that this is somehow less "formal" than something involving a bunch of funny symbols is a common misapprehension.