4.3 Geometric distribution  (Page 4/14)

References

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“Millennials: Confident. Connected. Open to Change.” Executive Summary by PewResearch Social&Demographic Trends, 2013. Available online at http://www.pewsocialtrends.org/2010/02/24/millennials-confident-connected-open-to-change/ (accessed May 15, 2013).

“Prevalence of HIV, total (% of populations ages 15-49),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/SH.DYN.AIDS.ZS?order=wbapi_data_value_2011+wbapi_data_value+wbapi_data_value-last&sort=desc (accessed May 15, 2013).

Pryor, John H., Linda DeAngelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman: National Norms Fall 2011. Los Angeles: Cooperative Institutional Research Program at the Higher Education Research Institute at UCLA, 2011. Also available online at http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf (accessed May 15, 2013).

“Summary of the National Risk and Vulnerability Assessment 2007/8: A profile of Afghanistan,” The European Union and ICON-Institute. Available online at http://ec.europa.eu/europeaid/where/asia/documents/afgh_brochure_summary_en.pdf (accessed May 15, 2013).

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“UNICEF reports on Female Literacy Centers in Afghanistan established to teach women and girls basic resading [sic] and writing skills,” UNICEF Television. Video available online at http://www.unicefusa.org/assets/video/afghan-female-literacy-centers.html (accessed May 15, 2013).

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 = $1-p$ where x = number of trials until first success for the formula $P(X=x)={(1-p)}^{x-1}p$ where the number of trials is up and including the first success.

An alternative formulation of the geometric distribution asks the question: what is the probability of x failures until the first success? In this formulation the trial that resulted in the first success is not counted. The formula for this presentation of the geometric is:

Formula review

$P(X=x)=p(1-p{)}^{x-1}$

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 = $\frac{1}{p}$ .

The standard deviation is σ = $\sqrt{\frac{1–p}{p^2}}$ = $\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}$ .

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. 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.