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I want to write formally the following sentence:

The sequence $S$ contains elements of the set $A$. Probability value $P(a)$ for element $a$ is defined as a number of its occurences in the sequence $S$, divided by count of all its elements.

I can write it the following way:

$$ S = (s_1, s_2, ..., s_n) : s_i \in A $$ $$ P(a) = {{ \left| \lbrace i \in \lbrace 1, 2, ..., n \rbrace : s_{i} = a \rbrace \right| } \over {n}}, \text{ given } n > 0\text{ and }a \in A $$

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...

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2 Answers

up vote 2 down vote accepted
  1. 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.

  2. 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.

  3. Most straightforward would be to write

    Let $C(x)$ be the number of elements of $s_1,\ldots,s_n$ that are equal to $x$. Then…

    The idea that this is somehow less "formal" than something involving a bunch of funny symbols is a common misapprehension.

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You know, I guess, that your third solution will suit my needs. After all, my priority is that someone understands my notation - writing the sentences as formally as possible is not really as important. –  Spook Mar 11 '13 at 7:46
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It's also not quite correct. How about $|\{i\in\{1,\ldots,n\}\colon s_i=a\}|$ in the numerator? If you view the sequence $S$ as a function $S\colon\{1,\ldots,n\}\to A$, $i\mapsto s_i$, then you might even write $|S^{-1}(a)|$ for the numerator. And the denominator should simply be $n$ (which you implicitly defined for $S$). As $S$ is not (primarily) a set, $|S|$ looks strange.

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You're right about my notation, I corrected it. However, I don't like the solution with $S^{-1}$ much, that looks like redefining the sequence to simplify the notation. I thought rather of a mathematical construct corresponding to a 'count of' function. Isn't there one? –  Spook Mar 10 '13 at 15:55
    
@Spook Actually, that normally isn't really redefining the ntotion of sequence. But nevertheless, isn't $|\{\ldots\}|$ what corresponds to a count function after all? All that matters is that you can define your frequency notion clear and unambiguously, not necessarily with less then five symbols ... –  Hagen von Eitzen Mar 11 '13 at 7:41
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