1. 1 Bernoulli and Binomial Distributions

2. 2 Bernoulli Random Variables Setting: –finite population –each subject has a categorical response with one of 2 possible values (0/1) –pick a simple random sample of n=1 subject Y random variable representing response (a Bernoulli random variable) Prob(Y=1)

3. 3 Bernoulli Random Variables Example: Finite population of 100 subjects, where 40 are normal weight and 60 are overweight. Response: 0 normal weight 1 overweight Population Parameters: Mean Variance

4. 4 Bernoulli Random Variables Example: Finite population of 100 subjects, where 40 are normal weight and 60 are overweight. Values: 0 normal weight 1 overweight Population Parameters: Mean Variance Pick a single subject at random: a Bernoulli Random Variable

5. 5 Bernoulli Random Variables Example: Finite population of 100 subjects, where 40 are normal weight and 60 are overweight. Values: 0 normal weight 1 overweight Probability a Bernoulli Random Variable EventyP(y) Normal01-p Overwt1p Total 1 events are mutually exclusive exhaustive probabilities sum to 1

6. 6 Bernoulli Random Variables Example: Finite population of 100 subjects, where 40 are normal weight and 60 are overweight. Values: 0 normal weight 1 overweight a Bernoulli Random Variable EventyP(y) Normal01-p Overwt1p Total 1

7. 7 Bernoulli Random Variables Example: Finite population of 100 subjects, where 40 are normal weight and 60 are overweight. Values: 0 normal weight 1 overweight a Bernoulli Random Variable Simple random sample of n=1

8. 8 Binomial Random Variable Binomial Random Variable: The sum of independent identically distributed Bernoulli random variables. Example: Finite population of 100 subjects, where 40 are normal weight and 60 are overweight. Values: 0 normal weight 1 overweight Select a simple random sample of size n with replacement –the random variable representing each selection is a Bernoulli Random variables –the random variables are independent –the random variables are identically distributed iid = independent and identically distributed (always occurs for random variables representing selections using simple random sampling with replacement) a Binomial Random Variable

9. 9 Independent Variables Are the two random variables independent? first selection in a sample second selection in a sample (with Rep) Two random variables are independent if for any realized value of the first random variable, the probability is unchanged for any realized value of the second random variable.

10. 10 Independent Variables Are the two random variables independent? first selection in a sample second selection in a sample (with Rep) Suppose Conclusion: The RV’s are independent

11. 11 Independent Variables Are the two random variables independent? first selection in a sample second selection in a sample (without Rep) a Bernoulli Random Variable

12. 12 Independent Variables Are the two random variables independent? first selection in a sample second selection in a sample (without Rep) Suppose Conclusion: The RV’s are not independent

13. 13 Binomial Random Variable Binomial Random Variable: The sum of independent identically distributed (iid) Bernoulli random variables. a Binomial Random Variable a vector of Random Variables

14. 14 Expected Value and Variance of a Vector of Random Variables a vector of Random Variables

15. 15 Expected Value and Variance of a Vector of Random Variables a vector of independent Random Variables a vector of independent and identically distributed (iid)Random Variables zero covariances identity matrix

16. 16 Expected Value and Variance of a Linear Combination of Random Variables a Binomial Random Variable a vector of independent and identically distributed Bernoulli Random Variables In general

17. 17 Variance of a Binomal Random Variables

18. 18 Expected Value and Variance of a Binomal Random Variable a Binomial Random Variable a vector of independent and identically distributed Bernoulli Random Variables

19. 19 Binomial Distribution see table A.1 in Appendix of Text nk=X0.4=p X=x 400.1785 10.3456 2 30.1536 40.0256

20. 20 Binomial Distribution see table A.1 in Appendix of Text nk0.6 440.1785 30.3456 2 10.1536 00.0256

21. 21 SRS with rep: Seasons Study With Seasons Study, define High Total Cholesterol: TC>240 Select SRS with replacement: Run SAS program: ejs09b540p46.sas Example: Change Program to get 5 samples of size n=10 For each, calculate total TC>240

22. 22 Binomial Distribution What if 10,000 Samples were selected?

23. 23 Binomial Distribution P(X=x=# with TC>240)= =(# ways of ways of picking samples with x)Pr(x ‘success’)P(n-x ‘failures’)

24. 24 Binomial Distribution Likelihood We select a srs with replacement of n=10 and observe x=4. What is p? This is a function of p

25. 25 Binomial Distribution Likelihood We select a srs with replacement of n=10 and observe x=4. What is p? Likelihood: Use table to find values for p: pL(p)p 0.050.0010.400.2508 0.100.01120.450.2384 0.150.04010.500.2051 0.200.08810.550.1596 0.250.14600.600.1115 0.300.20010.650.0689 0.350.2377etc

26. 26 Binomial Distribution Maximum Likelihood Likelihood: pL(p)p 0.050.0010.400.2508 0.100.01120.450.2384 0.150.04010.500.2051 0.200.08810.550.1596 0.250.14600.600.1115 0.300.20010.650.0689 0.350.2377etc 0.05 0.1 0.2 0.30.40.5 Maximum Likelihood

27. 27 Binomial Distribution- Differences in Use Mean Usually report “total” instead of “mean”. Total Estimate Variance Estimated Variance Use Normal CLT

28. 28 Binomial Distribution- Differences in Use MeanTotal Use Normal Dist for Interval Estimates Approximation good when and

29. 29 Binomial Distribution- Differences in Use Use hypothesized p for variance when and

30. 30 Binomial Distribution- CI for Difference in Prop. Diff in Means (Proportions see 14.6)

31. 31 Binomial Distribution- Hyp. Test for Difference in Prop. Pooled prob

32. 32 Chi-Square Distribution Hyp. Test for Difference in Prop. Under the null hypothesis, this statistic follows a chi-square distribution with 1 degree of freedom.