Chapter 5.4: Bernoulli and Binomial Distribution Chris Morgan, MATH G160 csmorgan@purdue.edu February 6, 2012 Lecture 12

Bernoulli Distribution Some problems in probability involve observing whether a specified event occurs. Noted as a “success” or a “failure” Examples: –Probability of a head in a single coin flip –Probability of having a boy –Probability of getting a raise

Bernoulli Distribution Notation: X ~ Bernoulli(p) or X ~ Ber(p) where p is a probability, and the PMF is given by: Expectation and Variance of the Bernoulli Distribution: E(X) = 0 * (1-p) + (1 * p) = p E(X²)= 0² * (1-p) + 1² * p = p Var(X) = p - p² = p(1-p) X 1 p(x) 1-p p

Example #1 Suppose our class passed (C or better) the last exam with probability 0.75. Let the random variable X be the probability that someone passes the exam. X ~ Ber(0.75) with PMF: E(X) = p = 0.75 Var(X) = p(1-p) = (0.75)(0.25) = 0.1875 X 1 p(x) 1-.75 .75

Binomial Distribution Obviously, a Bernoulli trial is quite boring and not practical at all. What if a Bernoulli event happens many times? We call that a Binomial event. We have done some of these problems before Binomial distribution only has a “success” or “failure” outcome, but it is repeated many times

Binomial Distribution Examples: Probability of getting 7 out of 40 heads Probability the Minnesota Twins win 3 out of 4 games in their series with the Yankees. Probability JJ makes 9 out of 10 free throws. Any time there are only two outcomes, n times, we have a binomial distribution…

Binomial Distribution Notation: X ~ Bin(n,p) PMF: p(x) = Expectation: Variance: E(X) = np Var(X) = n*p*(1-p)

Binomial Example #1 In a study of brand recognition, 95% of consumers recognized Coke. The company randomly selects 4 consumers for a taste test. •Let X be the number of consumers who recognize Coke. •Write out the PMF table for this. X ~ Bin(n=4, p=0.95)

Binomial Example #1 X p(x) 1 2 3 4 X ~ Bin(n=4, p=0.95) P(X ≥ 1) =   1 2 3 4

Binomial Example #1 X ~ Bin(n=4, p=0.95) Calculate E(X): Or using the known formula: E(X) = np = 4(.95) = 3.8

Binomial Example #1 X ~ Bin(n=4, p=0.95) Calculate Var(X): Var(X) = E(X2) – [E(X)] 2 Or using the known formula: Var(X) = np(1-p) = 4(0.95)(0.05) = 0.19

Binomial Example #2 To test for ESP, we have 4 cards. They will be shuffled and one randomly selected each time, and you are to guess which card is selected. This is repeated 10 times. Let R be the number of times you guess a card correctly. 1. What are the distribution and parameters of R? 2. If someone were just guessing, did NOT have ESP, how many would you expect them to get right out of 10 tries?

Binomial Example #2 3. What is the probability they guess half of them correctly? 4. What is the probability they get at least 3 correct?

Binomial Example #3 An insurance company receives claims independently of one another for various incidents. Of these claims, 10% are claims for damages from vandalism. If the insurance company receives 20 claims, let V be the number of claims that are for vandalism. 1. What is the distribution and parameters of V? 2. What is the probability there are either 2 or 3 claims for vandalism?

Binomial Example #3 3. If there are 4 or more claims for vandalism, the company must send you a special report so you can have the police check out the situation. What is the probability you have to send the police? 4. Given you didn’t get a special report, what is the probability the company received exactly 3 claims for vandalism? 5. In 20 claims, how many do you expect to be claims for vandalism? 6. What is the variance of number of vandalism claims received? 7. If 5 agents are filing claims (each 20 have 20 claims), how many total claims for vandalism do you anticipate?

Binomial Example #4 On a multiple choice exam with 3 possible answers for each of 5 questions, what is the probability a student would get 4 or more answers correct just by guessing?

Binomial Example #5 It is believed that 20% of Americans do not have health insurance. Suppose this in true and let X equal the number with no health insurance in a random sample of n = 15 Americans. a. How is X distributed? b. Give the mean, variance, and standard deviation of X. c. Find P(X ≥ 5)

Binomial Example #6 It is claimed that 15% of ducks in a particular region have patent schistosome infection. Suppose that seven ducks are selected at random. Let X equal the number of dicks that are infected. a) Assuming independence, how is X distributed? b) Find P(X > 2) c) Find P(X = 1) d) Find P(X < 5)

Binomial Example #7 A newsboy purchases papers at 10 cents and sells them at 15 cents. However, he is not allowed to return unsold papers. If his daily demand is a binomial random variable with n = 10 and p = 1/3, approximately how many papers should he purchase so as to maximize his expected profit?

Binomial Example #8 Suppose that 200 points are selected independently and at random from the following square: {(x,y): -1≤x≤1, -1≤y≤1 }. Let W equal the number of number of points that fall in A = {(x,y): x2 + y2 < 1}. a) How is W distributed? b) Give the mean, variance, and standard deviation of W. c) What is the expected value of W/50?

Binomial Example #9 Suppose that when in flight, airplane engines will fail with probability 1-p independently from engine to engine. If an airplane needs a majority of its engines operative to make a successful flight, for what vales of p is 5-engine plane preferable to a 3-engine plane?

Binomial Example #10 A student is getting ready to take an important oral exam and is concerned about the possibility of having an "on" day or an "off" day. He figures that if he has an on day, then each of his examiners will pass him independently of each other with probability .8, whereas, if he has an off day, this probability will be reduced to .4. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student feels that he is twice as likely to have an off day as he is to have an on day, should he request an examination with 3 examiners or with 5 examiners?