Niemi Consulting

Yesterday we introduced the idea of a random variable. Today we discuss the easiest random variable to understand, the Bernoulli random variables. Basically any time an outcome has one of two possibilities, it can be modeled as a Bernoulli random variable. Some examples include:

• will the Minnesota Twins win the next World Series
• will the Democrats win the next presidency
• will the Koala bear become extinct in the next 10 years
• will humans set foot on Mars in the next 30 years

Each of these questions have two possible answers which is the key to identifying outcomes that can be modeled as Bernoulli random variables. Of course, the probabilities of the positive outcomes in the above examples are not equal. Therefore we need a parameter that will define how likely the outcome is. In this case, we call the parameter p since it represents the probability of an outcome. If Y is the outcome of one of the examples above then we write

In English, this notation says `Y is a Bernoulli random variable and the probability that Y is equal to 1 is p and the probability that Y is equal to 0 is 1-p’. For example, if the Twins have a 5% chance of winning the next world series and if Y=1 corresponds to the Twins winning, then we have Y~Ber(0.05).

The last concept we need to introduce in this post is the concept of a probability mass function or pmf. The pmf defines the probability for each outcome of our random variable. For a Bernoulli random variable the pmf is

The two possible outcomes for a Bernoulli random variable are 0 and 1. If we plug these into the pmf, we find that the probability of Y=1 is p, written P(Y=1)=p, and the probability of Y=0 is 1-p, written P(Y=0)=1-p.

Bottom Line: A Bernoulli random variable can be used to model any event that has two exclusive outcomes.