Discrete Random Variables

Discrete variables are usually within certain limits. Examples: scores on a die, number of goals a football team scores, shoe size, height of a person - we learnt this earlier. On this page, we want to look at some techniques you can use to create models to describe this data. If you are looking at a discrete model say, how many passengers in a car, we could say that it is a random variable because it's at random - there isn't a way you can predict without being given a probability. Discrete random variables may have a finite or infinite number of outcomes. In the car example, we can say that it's finite as the maximum people in a car is usually five, but then again the maximum could be eight, depending on the size. So we could say that there are eight possible outcomes.
Another example could be the amount of people visiting this website each day - there is no maximum, it's infinite.

Notation and Conditions for a Discrete Random Variable
We usually denote a discrete random variable with an upper case letter e.g. X. Then the values that the variable can take are denoted by lower case letters, e.g. r. They could also be given suffixes for each value of r, e.g. r1. Therefore, we would say that the probability of X taking the value of r1 would be written out like this: P(X = r1. P(X = r) would probably be more suitable to a table heading for example. A shorter way of writing different probabilities are p1, p2 etc. If we had a finite amount of outcomes with a maximum of n, we could say the outcomes are r1, r2, r3... rn with the probabilities p1, p2... pn. The sum of all probabilities have to equal 1. We can then say that all possibilities are covers and they are exhaustive. We can also write this as:

n∑k = 1 = n∑k = 1 P (X = rk = 1.


Diagrams of Discrete Random Variables
The most appropriate diagram for discrete random variables is a vertical line chart.






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