Binomial Distribution

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Presentation transcript:

Binomial Distribution The binomial distribution is a discrete distribution.

Binomial Experiment A binomial experiment has the following properties: experiment consists of n identical and independent trials each trial results in one of two outcomes: success or failure P(success) = p P(failure) = q = 1 - p for all trials The random variable of interest, X, is the number of successes in the n trials. X has a binomial distribution with parameters n and p

EXAMPLES A coin is flipped 10 times. Success = head. X = n = p = Twelve pregnant women selected at random, take a home pregnancy test. Success = test says pregnant. X = n = p = ? Random guessing on a multiple choice exam. 25 questions. 4 answers per question. Success = right answer.

Examples when assumptions do not hold Basketball player shoots ten free throws Feedback affects independence and constant p Barrel contains 3 red apples and 4 green apples; select 4 apples without replacement; X = # of red apples. Without replacement implies dependence

What is P(x) for binomial?

Mean and Standard Deviation The mean (expected value) of a binomial random variable is The standard deviation of a binomial random variable is

Example Random Guessing; n = 100 questions. Expected Value = Probability of correct guess; p = 1/4 Probability of wrong guess; q = 3/4 Expected Value = On average, you will get 25 right. Standard Deviation =

Example Cancer Treatment; n = 20 patients Probability of successful treatments; p = 0.7 Probability of no success; q = ? Calculate the mean and standard deviation.

Normal Approximation For large n, the binomial distribution can be approximated by the normal, is approximately standard normal for large n.