Chapter 4: Discrete Random Variables
4.4 Geometric Distribution
Learning Objectives
By the end of this section, you should be able to:
- Identify the components of a geometric experiment
- Use the formulas for a geometric random variable to compute mean, variance, and standard deviation
There are three main characteristics of a geometric experiment.
- There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating what you are doing until the first success. Then you stop. For example, you throw a dart at a bullseye until you hit the bullseye. The first time you hit the bullseye is a “success” so you stop throwing the dart. It might take six tries until you hit the bullseye. You can think of the trials as failure, failure, failure, failure, failure, success, STOP.
- In theory, the number of trials could go on forever. There must be at least one trial.
- The probability [latex]p[/latex] of a success and the probability [latex]q[/latex] of a failure is the same for each trial with [latex]p+q = 1[/latex], so [latex]q = 1-p[/latex].
The geometric random variable [latex]X =[/latex] the number of independent trials until the first success. The probability that [latex]X[/latex] equals a particular value [latex]x[/latex], that is, the probability we first succeed on a particular trial, is
[latex]P(X = x) = q^{(x-1)} p.[/latex]
Example
Suppose that when rolling a fair die, you want to know the probability of getting the first three on the fifth roll.
The experiment here is to roll a fair die until you get a three, then stop.
So for the event to occur, on rolls one through four, you do not get a face with a three. The probability for each of the rolls is [latex]q = \frac{\text{5}}{\text{6}}[/latex], the probability of a failure. The probability of getting a three on the fifth roll is [latex]\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)\left(\frac{1}{6}\right)[/latex] = 0.0804
Example
A safety engineer feels that 35% of all industrial accidents in her plant are caused by failure of employees to follow instructions. She decides to look at the accident reports (selected randomly and replaced in the pile after reading) until she finds one that shows an accident caused by failure of employees to follow instructions. On average, how many reports would the safety engineer expect to look at until she finds a report showing an accident caused by employee failure to follow instructions? What is the probability that the safety engineer will have to examine at least three reports until she finds a report showing an accident caused by employee failure to follow instructions?
Let X = the number of accidents the safety engineer must examine until she finds a report showing an accident caused by employee failure to follow instructions. X takes on the values 1, 2, 3, …. The first question asks you to find the expected value or the mean. The second questio