The points of a sample space may be very concrete objects. For example, on tossing a coin, a head or tail is realized, on tossing a coin twice, a head-head, head-tail, tail-head or tail-tail is realized, on rolling die, one of the numbers \(1,2,3,4,5,6\) is realized, on drawing a card from a standard deck of cards, one of four suits – club, spade, heart, diamond and one of \(13\) ranks is realized.

However, sometimes the sample space of an experiment can be incredibly complicated or higher dimensional and the outcomes may be non-numeric. For example, in a sequence of \(N\) coin tosses, the sample space has an enormous \(2^N\) sample points. In a survey of a random sample of people in a city, various questions may have numeric (e.g. age or height) and non-numeric answers (e.g. favorite political party or favorite movie). Thus, we require a mathematical description of the sample points.

Random variables fulfill this purpose. They assign numerical values to each sample point.

Random variables and their distributions