Bernoulli

1. Bernoulli Bernoulli population is a population in which each element is one of two possibilities. The two possibilities are usually designed as success and failure. The probability of success is p which means that the probability of failure is 1-p. A Bernoulli trial is observing one element in a Bernoulli population. Bernoulli population is a population in which each element is one of two possibilities. The two possibilities are usually designed as success and failure.Bernoulli population is a population in which each element is one of two possibilities. The two possibilities are usually designed as success and failure.

2. Example Ex1. The toss of a coin results in a Bernoulli population. Ex2. The homes in Dallas either heat with gas or not. Choices to heat with gas or not is a Bernoulli population. Ex3. In a local election, the voters either favor a candidate for mayor or do not. The choices of voters result in a Bernoulli population.

3. Binomial experiment A Binomial experiment is an experiment that consists of n repeated independent Bernoulli trials in which the probability of success on each trial is ? and the probability of failure on each trial is 1- ? .

4. Criteria for a Binomial Setting 1.consists of n Bernoulli trials (each trial yields either S or F. 2.For each trial, P(S)= ? ,and P(F)=1- ? . 3.The trials are independent. Ex1. The toss of a coin 3 times Fixed number of trials. The outcome for each trial must be independent of the others. The probability of success is the same for each trial. There are only 2 possible outcomes for each trial � success or failure. Fixed number of trials. The outcome for each trial must be independent of the others. The probability of success is the same for each trial. There are only 2 possible outcomes for each trial � success or failure. Fixed number of trials. The outcome for each trial must be independent of the others. The probability of success is the same for each trial. There are only 2 possible outcomes for each trial � success or failure. Fixed number of trials. The outcome for each trial must be independent of the others. The probability of success is the same for each trial. There are only 2 possible outcomes for each trial � success or failure.

5. binomial random variable. The random variable x, which gives the number of success in the n trials of Bernoulli experiment, is called a binomial random variable. The sample space of x is  Sx = {0, 1, 2, � , n}. Ex1. The number of head after tossing a coin 10 times.

6. Question: Are the following random variables binomial random variable? The number of getting head after flipping a coin 10 times. 40% of all airline pilot are over 40 years of age. The number of pilots who are over 40 out of 15 randomly chosen pilots. Suppose a salesperson makes sales to 20% of her customers, the number of customers until her first sale. A room contains 6 women and four men, Three people are selected to form a committee. The number of women on the committee. Once a random variable is identified as being a binomial random variable, we can determine its probability distribution. The salesperson may never make a sale, hence the sample space for x is S_x={0, 1,�} which clearly is not the sample space of a binomial random variable. If x had been defined as # of sales out of, say, 50 customers, then x would have been a binomial random Once a random variable is identified as being a binomial random variable, we can determine its probability distribution. The salesperson may never make a sale, hence the sample space for x is S_x={0, 1,�} which clearly is not the sample space of a binomial random variable. If x had been defined as # of sales out of, say, 50 customers, then x would have been a binomial random

7. Suppose a couple plans to have 3 children. The chance they have a boy is 0.2. The gender of one child is independent of the gender of another child. Let X be the number of boys they have.

8. If the chance a child is a boy is 0.2, what�s the chance a child is a girl? How many gender sequences (i.e. BBB, BBG, BGG, etc) are possible?

9. We want to fill in the probability distributionn below:

10. P(X = 1) = P(only one boy) = P(GGB) + P(GBG) + P(BGG) = [(0.8)2(0.2)] + [(0.8)2(0.2)] + [(0.8)2(0.2)] = 3 (0.8)2(0.2)

11. Similarly, P(X=2) equals 3(0.2)2(0.8). Now we can complete the probability distribution of X.

12. Binomial distribution Generally, suppose there are n trials, with ? being the probability of success on each trials, for there to be exact k success, there must be (n-k) failure. Moreover, 1) There are different combinations for the k successes. 2) The probability of a given arrangement of k successes and n-k failures is ?(1-?) . Thus, the probability of k successes in n trials isGenerally, suppose there are n trials, with ? being the probability of success on each trials, for there to be exact k success, there must be (n-k) failure. Moreover, 1) There are different combinations for the k successes. 2) The probability of a given arrangement of k successes and n-k failures is ?(1-?) . Thus, the probability of k successes in n trials is

13. Ex: 50% of homes in Dallas are heated with gas. Let Y = # of homes out of 6 that are heated with gas. The sample space of the value of y is Sy = {0, 1, 2, 3, 4, 5, 6} It is a Binomial random variable. What is the distribution of Y?

14. Solution When n is very large, computing the probabilities becomes very tedious. In fact, We can use computer or binomial table to find out the probabilities for certain values of n, k and ? . Look at table B.1 on Appendix B.

15. How the two parameters (n, ?) affect binomial distribution? Fix n, ? is more close to .5, the distribution is more near to a bell curve. If ? is more close to 0, the distribution is more skewed to the right. If ? is more close to 1, the distribution is more skewed to the left.

17. When the success probability ? on each trial is fixed and n is lager, the distribution is more close to a bell curve

19. mean and standard deviation For binomial distribution Mean� = n ? Variances 2 = n? (1- ? ) Standard deviation s= It is apparent that the values of n and ? are very important because they determine the various binomial distribution. They are the parameters of the binomial distribution. It is apparent that the values of n and ? are very important because they determine the various binomial distribution. They are the parameters of the binomial distribution.

20. Example 4.25(page 274) Suppose that the probability of successfully rehabilitating a convicted criminal in a penal institution is .4. Let r represent the number successfully rehabilitated out of a random sample of ten convicted criminal, find out the mean and the standard deviation of the variable r.

21. 30% of homes in Dallas are heated with gas. Let Y = # of homes out of 6 that are heated with gas. What is the probability of {Y<3}? Example 4.13

22. 0 0.117649 1 0.302526 2 0.324135 P(Y<3)=0.74431

23. Probability of event in the Binomial case Calculate the unknown probability of random variable y with n=4 and ?=.2 P(y=2) = P(y<2)= P(y=2)= P(y>3)=