Probability Distributions On Countable Sets
Why
Suppose we flip a coin until it lands heads. What is the probability that we see a head after one flip? Two flips? $n$ flips? We want to extend our notion of probability distribution to a set with infinite elements, but only countably many.
Discussion
Consider a set $A$.1 If $A$ has $n$ elements, then a probability distribution on $A$ is $p: A \to \R $ where $p(a) = 1/n$. There is a natural candidate.
What if $A$ is the set of natural numbers $\N $. The principle difficulty is that not all sequences of real numbers $a: \N \to \R $ are summable.
Definition
A (countable) probability
distribution on $\N $ is $p: \N \to
\R $ where $p \geq 0$ and
\[
\textstyle
\sum_{n =1}^{\infty} p(n) = 1.
\]
More generally, a probability distribution on
some countable set $C$ which we have numbered
$c: \N \to C$, is a function $p: C \to
[0,1]$ such that
\[
\sum_{i = 1}^{n} p(c_i) = 1
\]
- Future editions will rework this sheet. ↩︎