Binomial Distribution The binomial distribution is a discrete

Binomial Distribution The binomial distribution is a discrete distribution.

Binomial Experiment w A binomial experiment has the following properties: n n experiment consists of n identical and independent trials each trial results in one of two outcomes: success or failure l P(success) = p l 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 w A coin is flipped 10 times. Success = head. n X = n = p = w Twelve pregnant women selected at random, take a home pregnancy test. Success = test says pregnant. n X = n = p = ? w Random guessing on a multiple choice exam. 25 questions. 4 answers per question. Success = right answer. n X = n = p =

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

What is P(x) for binomial?

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

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

Example w Cancer Treatment; n = 20 patients w Probability of successful treatments; p = 0. 7 w Probability of no success; q = ? w Calculate the mean and standard deviation.

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