6A Binomial and Poisson - PowerPoint Presentation

1. 6ABinomial and Poisson mean and variance

2. Binomial Distribution KUS objectivesBAT use formulas to find the mean and variance of the binomial distribution Starter: A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.39 of finding oil, independently of all other sites. The random variable X represents the number of sites investigated. The probability distribution of X is shown below.Find the expectation and variance of X. b) It costs £66000 to investigate each site. Find the expected total cost of the investigation.r1234P(X = r)0.390.23790.14510.227

3. Starter Solution

4. Notes derivation of Mean and Variance of a Binomial Distibution If X and Y are two independent random variables thenAnd  So And  Consider the random variable and the random variables which represent the number of successes on the nth trial. The distribution of each of the will be the same  It is easy to calculate = And = =  X is the total number of successes in the n trials and so  So you can apply the above result to get    

5. Notes  If random variable  

6. A fair cubical die is thrown 36 times and the random variable X represents the number of sixes obtained Find the mean and variance of XFind  a) The random variable  random variable    b) =          WB 21 mean and variance Checking this result with a calculator n x = 3 n = 36 p = 1/6 gives = 0.1277  

7. WB 22 mean and variance A fair, four sided die has the numbers 1, 2, 3 and 4 on its faces. The die is rolled 20 times. The random variable X represents the number of 4s obtained Find the mean and variance of XFind   a) The random variable      b) =        Use c 

8. The probability of a sales representative making a sale on a call is 0.15. Representatives are required to achieve a mean of 5 sales each day. Find the least number of calls each day a representative should make to achieve this targetThe random variable and mean = 0.15n  WB 23 mean and variance Let X = number of sales each dayAt least 34 calls

9. WB 24 solve inequality David believes that 35% of people in a certain town will vote for him in the next election and he commissions a survey to verify this. Find the minimum number of people the survey should ask to have a mean number of more than 100 voting for David. Let n = the number of people askedLet X = the number of people (out of n) voting for David    So  So the minimum number of people that should be asked is 286 

10. WB 25 solve inequality An examiner is trying to design a multiple choice test. For students answering the test at random, he requires that the mean score on the test should be 20 and the standard deviation should be at least 4. Assuming that each question has the same number of alternative answers, find how many questions and how many alternative answers each question should have. The number of alternatives for each question should be as few as possible. Let n = the number of questionsLet p = the probability of guessing a correct answerLet X = the number of questions answered correctly at random   So   So     …  So the examiner should use i.e. have 5 alternatives for each question And with gives i.e have 100 questions  

11. Practice Ex 1D

12. self-assess One thing learned is – One thing to improve is – KUS objectivesBAT use formulas to find the mean and variance of the binomial distribution

13. END