1. Today Today: Chapter 9 Assignment: 9.2, 9.4, 9.42 (Geo(p)=“geometric distribution”), 9-R9(a,b) Recommended Questions: 9.1, 9.8, 9.20, 9.23, 9.25

2. Estimation Can use the sample mean and sample variance to estimate the population mean and variance respectively How do we estimate parameters in general? Will consider 2 procedures: –Method of moments –Maximum likelihood

3. Method of Moments Suppose X=(X 1, X 2,…,X n ) represents random sample from a population Suppose distribution of interest has k parameters The procedure for obtaining the k estimators has 3 steps: –Conpute the first k population moments first moment is the mean, second is the variance, … –Set the sample estimates of these moments equal to the population moment –Solve for the population parameters

4. Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a population Suppose the population is Poisson Find the method of moments estimator for the rate parameter

5. Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a population with pdf The mean and variance of X are: Find the method of moments estimator for the parameter

6. Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a population with pdf The mean and variance of X are: Find the method of moments estimator for the parameters

7. Maximum Likelihood Suppose X=(X 1, X 2,…,X n ) represents random sample from a Ber(p) population What is the distribution of the count of the number of successes What is the likelihood for the data

8. Example Suppose X=(X 1, X 2,…,X 10 ) represents random sample from a Ber(p) population Suppose 6 successes are observed What is the likelihood for the experiment If p=0.2, what is the probability of observing these data? If p=0.5, what is the probability of observing these data? If p=0.6, what is the probability of observing these data?

10. Maximum Likelihood Estimators Maximum likelihood estimators are those that result in the largest likelihood for the observed data More specifically, a maximum likelihood estimator (MLE) is: Since the log transformation is monotonically increasing, any value that maximizes the likelihood also maximizes the log likelihood

11. Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a population Suppose the population is Poisson Find the MLE for the rate parameter

12. Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a population Suppose the population has pdf Find the MLE for θ

13. Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a normal population (N(μ,σ 2 ) ) Find the MLE for μ and σ 2

14. Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a normal population (N(μ,σ 2 ) ) Find the MLE for μ and σ 2