The sample space (denoted ) is defined as the set of all possible distinct outcomes or “events” of an experiment. Using this perspective, a classic idea about ‘probability’ is
provided all points in are equally likely.
Definition Let be a discrete sample space. Then the probabilities are numbers attached to the such that two conditions hold:
Definition An event in a discrete sample space is a subset . If the event contains only one point, e.g. we call it a simple event. An event A made up of two or more simple events, e.g. is a compound event.
Definition The probability of an event is the sum of the probabilities for all the simple events that make up .
A random variable is a numerical-valued variable that represents the outcomes in an experiment or random process. Typically, random variables are denoted by an upper-case letter such as . The corresponding lower case letter is often reserved to refer to one of a number of possible values that the random variable can take on. For example, if a coin is tossed 3 times then
Definition A random variable is a function that assignes a real number to each point in a sample space .
One type of desired description of a random variable is a summary for how probability is distributed amongst the possible values a random variable can take on.
Definition The probability function of a random variable is the function
The pairs is collectively the probability distribution. All probability functions share two properties:
An alternative to the probability distribution for describing a probability model is the cumulative distribution function or simply the distribution function.
Definition The distribution function of a random variable is
All distribution functions share three properties: