Discrete Probability Space

probability-theory

Definition

Discrete Probability Space

A discrete probability space is a probability space whose sample space is finite or countably infinite. Writing for , it is determined by the point masses

Why Point Masses Determine Every Event

Because is countable, every event is a countable disjoint union of singleton outcomes:

Countable additivity then gives

Thus no additional probabilities need to be specified once the singleton masses are known.

Construction from Weights

Conversely, suppose numbers satisfy

Then

defines a unique probability measure on . Non-negativity follows from , normalisation follows by taking , and countable additivity follows because disjoint events contain disjoint collections of outcomes.

Contrast with a Discrete Random Variable

Warning

A discrete probability space has a finite or countably infinite underlying sample space . A discrete random variable instead has a finite or countable image . A discrete random variable may therefore be defined on an uncountable probability space.

Example

Tosses until the first head

Let

where outcome means that the first head occurs on toss . For a fair coin, assign

These masses are normalised because

The probability that the first head occurs on an even-numbered toss is