6. Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7

7. Binomial Probability Distribution Consider a sequence of independent events with only two possible outcomes called success (S) and failure (F) Example: outcome of treatment (cured/not cured) opinion (yes/no) S=yes, F=no gender (boy/girl) S=boy, F=girl Let p be the probability of S Consider n number of such independent events. Then the total no of success out of n such events is a random variable called Binomial r.v.

8. Binomial Probability Distribution Let p = probability of having a boy q = probability of having a girl Assume the outcome of the first child (boy or girl) does not affect the outcome of the second and subsequent children i.e. events are independent, we can multiply the probabilities together to get Pr(BBB) = p x p x p = Pr(3 boys) Pr(GGG) = q x q x q = Pr(No boy)

10. Binomial Probability For a 3-child family, 1 boy = BGG or GBG or GGB Pr(1 boy) = Pr(BGG)+Pr(GBG)+Pr(GGB) = pqq + qpq +qqp = 3pqq Similarly, Pr(2 boys) = Pr(BBG)+Pr(BGB)+Pr(GBB) = 3ppq

11. Example 1 Choose a group of 7 old individuals randomly from the population of 65-74 years old in US. Suppose 12.5% of the population in that age is diabetic. The total no. of persons out of the 7 selected who suffers from Diabetes has a binomial distribution. LetX = # diabetic Binomial( n = 7, p = 0.125)

12. Example 2 Past records indicate that 70% of patients responded to treatment. What is the probability that 16 out of the next 20 patients will respond to treatment? # patients responding ~ Bin( n = 20, p = 0.7) P(16 out of 20 responded) = 0.1304

14. The number of deaths attributable to typhoid fever follows a Poisson distribution at a rate of 4.6 deaths per year Y = # deaths in 6 months ~ Poisson(2.3) X = # deaths in 1 year ~ Poisson(4.6)