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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. Estimation. Can use the sample mean and sample variance to estimate the population mean and variance respectively

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Today

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  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=(X1, X2,…,Xn) 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=(X1, X2,…,Xn) 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=(X1, X2,…,Xn) 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=(X1, X2,…,Xn) 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=(X1, X2,…,Xn) 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=(X1, X2,…,X10) 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?

  9. 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

  10. Example • Suppose X=(X1, X2,…,Xn) represents random sample from a population • Suppose the population is Poisson • Find the MLE for the rate parameter

  11. Example • Suppose X=(X1, X2,…,Xn) represents random sample from a population • Suppose the population has pdf • Find the MLE for θ

  12. Example • Suppose X=(X1, X2,…,Xn) represents random sample from a normal population (N(μ,σ2) ) • Find the MLE for μ and σ2

  13. Example • Suppose X=(X1, X2,…,Xn) represents random sample from a normal population (N(μ,σ2) ) • Find the MLE for μ and σ2

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