A discrete random variable is one in which all the outcomes are mutually exclusive and cannot be broken down into more precise measurements. Discrete random variables exist on either a finitely or infinitely countable continuum. For example, when you flip a coin, the two possible outcomes, heads or tails, are mutually exclusive. You can't have part of a head and part of a tail on any given flip. Another example of a random variable is how many cars are on a street at any given time. There can be one car, two cars or 10 cars. However, there can't be three-quarters or one-third of a car.
When given a discrete random variable, Z, you can establish the probability of each outcome with a probability function. For example, if you flip a coin and z1=heads and z2=tails, the probability function is as follows: f(z)= {1/2 for Z=z1, 1/2 for Z=z2. The probability of each discrete event must add up to one. Probability functions for discrete random variables can be represented using a bar graph.
A continuous random variable is one that measures an infinitely uncountable probability space. As such, although each event is unique, the probability of any one event cannot be directly measured because it can always be further broken down into smaller parts. Height is an example of a continuous random variable because it can always be measured more precisely. If you measure someone in feet, they can be more precisely measured in inches. If measured in inches, a more precise measurement can be made in centimeters, then millimeters, then decimeters.
When given a continuous random variable, Z, you can calculate the probability of any range of events using a probability density function. Because the sample space is infinitely uncountable, it is not possible to measure any one event directly. For example, if you are measuring the probability of individuals being a certain height, the probability density function might look like this: f(z): { z for a<z<b, 0 otherwise. The probability function can be integrated to find the probability of any range of events. The integration of a probability density function, f(z), is known as a density function and can be denoted as F(z). For example, F(z)= ∫ f(z) dz. A function is only a probability density function if the integral of the function between the defined domain equals one.