Distributions

Random Variables

Technically speaking, a probability distribution is a representation of the probabilities of all possible outcomes of a random phenomenon. A random [Page 540]phenomenon can be an experiment or a measurement, for example. In this entry, the example of a random phenomenon will be to flip a coin twice and record each flip. The set of all possible outcomes of the random phenomenon is called the sample space. Flipping a coin twice yields the following sample space: S = {HH, HT, TH, TT}, where H stands for head and T for tail. Dependent on the information of the data one is interested in, a random variable assigns a number to each outcome of the sample space. For example, if we are interested in the number of heads in this random phenomenon, we get a random variable that has a 2 for the outcome in which we observe HH, a 1 for both outcomes HT and TH, and a 0 for TT. Another example for a random variable could be to record if the first flip is a head (1) or not (0). Then we would assign a 1 to the outcome HH and HT and a 0 to TH and TT.

If a random variable is discrete, any outcome can have a natural number assigned to it. No further number can be added between these two. For example, a random variable that indicates how many items of an exam were answered correctly is discrete.

A random variable is called continuous whenever there are theoretically an infinite number of values between any two values. Therefore, it is not possible to assign a natural number to any possible outcome. Here, decimal numbers are used. One example of a continuous random variable would be if the time a person takes to complete an exam was measured.

In the coin flipping experiment, the random variables are discrete. One example for a continuous random variable for this example would be the amount of time it takes to flip a coin until head appears once.

X can be used to represent the random variable, while x represents a particular value of the variable.