Binomial Distribution

1. Binomial Distribution The binomial distribution is a discrete distribution.

2. 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

3. 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. • X = n = p =

4. 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

5. What is P(x) for binomial?

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

7. Example • Random Guessing; n = 100 questions. • 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 =

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

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