# Discrete Random Variables

## Geometric Distribution (Optional)

There are three main characteristics of a geometric experiment:
1. Repeating independent Bernoulli trials until a success is obtained. Recall that a Bernoulli trial is a binomial experiment with number of trials n = 1. 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 bull's-eye until you hit the bull's-eye. The first time you hit the bull's-eye is a success so you stop throwing the dart. It might take six tries until you hit the bull's-eye. You can think of the trials as failure, failure, failure, failure, failure, success, stop.
2. In theory, the number of trials could go on forever. There must be at least one trial.
3. The probability, p, of a success and the probability, q, of a failure do not change from trial to trial. p + q = 1 and q = 1 − p. For example, the probability of rolling a three when you throw one fair die is 
 . This is true no matter how many times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls one through four, you do not get a face with a three. The probability for each of the rolls is q = 
 , the probability of a failure. The probability of getting a three on the fifth roll is 
  = .0804.
X = the number of independent trials until the first success.
p = the probability of a success, q = 1  p = the probability of a failure.
There are shortcut formulas for calculating mean μ, variance σ2, and standard deviation σ of a geometric probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book.

### Notation for the Geometric: G = Geometric Probability Distribution Function

X ~ G(p)
Read this as X is a random variable with a . The parameter is p; p = the probability of a success for each trial.

### References

Central Intelligence Agency. (n.d.). The world factbook. Retrieved from https://www.cia.gov/library/publications/theworld-factbook/geos/af.html
Pew Research Center. (n.d.). Millennials: A portrait of generation next. Retrieved from http://www.pewsocialtrends.org/files/2010/10/millennials-confident-connected-open-to-change.pdf
Pew Research. (2013). Millennials: confident. Executive Summary: Pew Research Social & Demographic Trends. Retrieved from http://www.pewsocialtrends.org/2010/02/24/millennials-confident-connected-open-tochange/
Pryor, J. H., et al. (2011). The American freshman: National norms fall 2011. Los Angeles: Cooperative Institutional Research Program, Higher Education Research Institute. Retrieved from http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/ TheAmericanFreshman2011.pdf
The European Union and ICON-Institute. (2007/8). Summary of the national risk and vulnerability assessment 2007/8: A profile of Afghanistan. Retrieved from http://ec.europa.eu/europeaid/where/asia/documents/afgh_brochure_summary_en.pdf
The World Bank. (2013). Prevalence of HIV, total (% of populations ages 15-49). Retrieved from http://data.worldbank.org/indicator/SH.DYN.AIDS.ZS?order=wbapi_data_value_2011+wbapi_data_value+wbapi_data_value-last&sort=desc
UNICEF Television. (n.d.). UNICEF reports on female literacy centers in Afghanistan established to teach women and girls basic reading and writing skills. (Video). Retrieved from http://www.unicefusa.org/assets/video/afghan-femaleliteracy-centers.html

### Chapter Review

There are three characteristics of a geometric experiment:
1. There are one or more Bernoulli trials with all failures except the last one, which is a success
2. In theory, the number of trials could go on forever; there must be at least one trial
3. The probability, p, of a success and the probability, q, of a failure are the same for each trial
In a geometric experiment, define the discrete random variable X as the number of independent trials until the first success. We say that X has a geometric distribution and write X ~ G(p) where p is the probability of success in a single trial.
The mean of the geometric distribution X ~ G(p) is μ = 
  = 
 .

### Formula Review

X ~ G(p) means that the discrete random variable X has a geometric probability distribution with probability of success in a single trial p.
X = the number of independent trials until the first success
X takes on the values x = 1, 2, 3, . . .
p = the probability of a success for any trial
q = the probability of a failure for any trial p + q = 1 : q = 1 – p
The mean is μ = 
 .
The standard deviation is σ = 
 
 = 
 
.
Use the following information to answer the next six exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agree with a recent law that was passed. Suppose that you randomly select freshman from the study until you find one who replies yes. You are interested in the number of freshmen you must ask.

### HOMEWORK
