# Discrete Random Variables

## Poisson Distribution (Optional)

There are two main characteristics of a Poisson experiment.
1. The Poisson probability distribution gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are five words spelled incorrectly in 100 pages. The interval is the 100 pages.
2. The Poisson distribution may be used to approximate the binomial if the probability of success is small (such as .01) and the number of trials is large (such as 1,000). You will verify the relationship in the homework exercises. n is the number of trials, and p is the probability of a success.
The random variable X = the number of occurrences in the interval of interest.

### Notation for the Poisson: P = Poisson Probability Distribution Function

X ~ P(μ)
Read this as X is a random variable with a Poisson distribution. The parameter is μ (or λ); μ (or λ) = the mean for the interval of interest.

### References

Centers for Disease Control and Prevention. (2012, Oct. 2). Teen drivers: Get the facts. Retrieved from http://www.cdc.gov/Motorvehiclesafety/Teen_Drivers/teendrivers_factsheet.html
Daily Mail. (2011, June 9). One born every minute: the maternity unit where mothers are THREE to a bed. Retrieved from http://www.dailymail.co.uk/news/article-2001422/Busiest-maternity-ward-planet-averages-60-babies-day-mothersbed.html
Department of Aviation at the Hartsfield-Jackson Atlanta International Airport. (2013). ATL fact sheet. Retrieved from http://www.atlanta-airport.com/Airport/ATL/ATL_FactSheet.aspx
Lenhart, A. (2012). Teens, smartphones & testing: Texting volume is up while the frequency of voice calling is down. About one in four teens say they own smartphones. Pew Internet. Retrieved from http://www.pewinternet.org/~/media/Files/Reports/2012/PIP_Teens_Smartphones_and_Texting.pdf
Ministry of Health, Labour, and Welfare. (n.d.). Children and childrearing. Retrieved from http://www.mhlw.go.jp/english/policy/children/children-childrearing/index.html
Pew Internet. (2013). How Americans use text messaging. Retrieved from http://pewinternet.org/Reports/2011/Cell-Phone-Texting-2011/Main-Report.aspx
South Carolina Department of Mental Health. (2006). Eating disorder statistics. Retrieved from http://www.state.sc.us/dmh/anorexia/statistics.htm
The Guardian. (2011, June 8). Giving birth in Manila: The maternity ward at the Dr Jose Fabella Memorial Hospital in Manila, the busiest in the Philippines, where there is an average of 60 births a day. Retrieved from http://www.theguardian.com/world/gallery/2011/jun/08/philippines-health#/?picture=375471900&index=2
Vanderkam, L. (2012, Oct. 8). Stop checking your email, now. CNNMoney. Retrieved from http://management.fortune.cnn.com/2012/10/08/stop-checking-your-email-now/
World Earthquakes. (2012). World earthquakes: Live earthquake news and highlights. Retrieved from http://www.worldearthquakes.com/index.php?option=ethq_prediction

### Chapter Review

A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time or space, if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial, if the probability of success is small (less than or equal to .05) and the number of trials is large (greater than or equal to 20).

### Formula Review

X ~ P(μ) means that X has a Poisson probability distribution where X = the number of occurrences in the interval of interest.
X takes on the values x = 0, 1, 2, 3, . . .
The mean μ is typically given.
The variance is σ2 = μ, and the standard deviation is 
 .
When P(μ) is used to approximate a binomial distribution, μ = np where n represents the number of independent trials and p represents the probability of success in a single trial.
Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.
Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.

### HOMEWORK

Use the following information to answer the next two exercises: The average number of times per week that Mrs. Plum’s cats wake her up at night because they want to play is 10. We are interested in the number of times her cats wake her up each week.