# Discrete Random Variables

## Binomial Distribution (Optional)

There are three characteristics of a binomial experiment:

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials.
There are shortcut formulas for calculating mean μ, variance σ2, and standard deviation σ of a binomial probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book.
Here n is the number of trials, p is the probability of a success, and q is the probability of a failure.

### Notation for the Binomial: B = Binomial Probability Distribution Function

X ~ B(n, p)
Read this as X is a random variable with a binomial distribution. The parameters are n and p: n = number of trials, p = probability of a success on each trial.
The following is the interpretation of the mean 
 
   and standard deviation 
 
  :
If you randomly select 20 adult workers, and do that over and over, you expect around eight adult workers out of 20 to have a high school diploma but do not pursue any further education on average. And you expect that to vary by about two workers on average.

### References

American Cancer Society. (2013). What are the key statistics about pancreatic cancer? Retrieved from http://www.cancer.org/cancer/pancreaticcancer/detailedguide/pancreatic-cancer-key-statistics
Central Intelligence Agency. (n.d.). The world factbook. Retrieved from https://www.cia.gov/library/publications/theworld-factbook/geos/af.html
ESPN NBA. (2013). NBA statistics  2013. Retrieved from http://espn.go.com/nba/statistics/_/seasontype/2
Newport, F. (2013). Americans still enjoy saving rather than spending: Few demographic differences seen in these views other than by income. GALLUP Economy. Retrieved from http://www.gallup.com/poll/162368/americansenjoy-saving-rather-spending.aspx
Pryor, J. H., et al. (2011). The American freshman: National norms fall 2011. Los Angeles, CA: Cooperative Institutional Research Program, Higher Education Research Institute. Retrieved from http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf
Wikipedia. (n.d.). Distance education. Retrieved from http://en.wikipedia.org/wiki/Distance_education
World Bank Group. (2013). Access to electricity (% of population). Retrieved from http://data.worldbank.org/indicator/EG.ELC.ACCS.ZS?order=wbapi_data_value_2009%20wbapi_data_value%20wbapi_data_value-first&sort=asc

### Chapter Review

A statistical experiment can be classified as a binomial experiment if the following conditions are met:
1. There are a fixed number of trials, n
2. There are only two possible outcomes, called success and failure, for each trial; the letter p denotes the probability of a success on one trial and q denotes the probability of a failure on one trial
3. The n trials are independent and are repeated using identical conditions
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. The mean of X can be calculated using the formula μ = np, and the standard deviation is given by the formula σ = 
 .

### Formula Review

X ~ B(n, p) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p.
X = the number of successes in n independent trials
n = the number of independent trials
X takes on the values x = 0, 1, 2, 3, . . . , n
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 of X is μ = np. The standard deviation of X is σ = 
 .
Use the following information to answer the next eight 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 agreed with a recent federal law that was passed.
Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number who agreed with that law.

### HOMEWORK

Use the following information to answer the next four exercises: Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4 percent. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu.
Use the following information to answer the next three exercises: The probability that a local hockey team will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games.
Let X = the number of games won in that upcoming month.