1. Binomial distribution Nutan S. Mishra Department of Mathematics and Statistics University of South Alabama

2. Binomial experiment An experiment is called binomial experiment if it satisfies following four conditions 1.Consists of n trials 2.All the trials are independent. 3.There are only two possible outcomes of each trial. 4.Probability of success in each trial is constant say p. X is number of successes in the experiment

3. Example Toss four fair coins. X= # heads showed up Then x may take values 0 or 1 or 2 or 3 or at the most 4. p =.5, n =4. Use binomial table to complete the following table xP(X) = 4 C x p x (1-p) 4-x 0 4 C 0 p 0 (1-p) 4 = 1 4 C 1 p 1 (1-p) 3 = 2 4 C 2 p 2 (1-p) 2 = 3 4 C 3 p 3 (1-p) 1 = 4 4 C 4 p 4 (1-p) 0 =

4. Example In a company it is known that among the population of employees 35% are smokers and 65% are non smokers. If we select a sample of size 10 employees from this population and count the number of smokers is the sample then X= # smokers in the sample of 10 is a binomial variable with n=10 and p=.35 that is (1-p) =.65. The possible values x takes : 0 to 10 Use table of combinations and a calculator to complete the table on next slide

5. X # smokers in the sample of size10 P(X)= 10 C x p x (1-p) 10-x 0 10 C 0 p 0 (1-p) 10 = 1 10 C 1 p 1 (1-p) 9 = 2 10 C 2 p 2 (1-p) 8 = 3 10 C 3 p 3 (1-p) 7 = 4 10 C 4 p 4 (1-p) 6 = 5 10 C 5 p 5 (1-p) 5 = 6 10 C 6 p 6 (1-p) 4 = 7 10 C 7 p 7 (1-p) 3 = 8 10 C 8 p 8 (1-p) 2 = 9 10 C 9 p 9 (1-p) 1 = 10 10 C 10 p 10 (1-p) 0 =