Today Today: Chapter 8 Assignment: 5-R11, 5-R16, 6-3, 6-5, 8-2, 8-8 Recommended Questions: 6-1, 6-2, 6-4, , 8-3, 8-5, 8-7 Reading: –Sections 8.1, 8.2, 8.7, 8.8, 8.10 –Will not do “suffiency” –Omit 8.4

Random Sampling A random sample of size n is a sequence of independent observations from the population Random sampling has a good chance of producing a representative sample (i.e., its accuracy reflects the population characteristic of interest) The joint p.d.f of the observations is:

Random Sampling We will only consider random samples from infinite distributions or with replacement

Example Suppose a random sample of size 5 is taken from a U(-1,1) distribution. What is the joint pdf of the data? Suppose a random sample of size 5 is taken from a N(5,9) distribution. What is the joint pdf of the data? Suppose a random sample of size n is taken from a N(μ,σ) distribution. What is the joint pdf of the data?

Models and Parameters In statistics, likelihood has a very specific meaning We shall deal mainly with models that have that are defined by a parameter, say, θ The probability model is written as f(x| θ)

Example Suppose a random sample of size 5 is taken from a N(5,9) distribution. What is the joint pdf of the data? The parameter that defines this model is:

Likelihood Let f(x| θ) be the joint pdf of the sample X=(X 1, X 2,…,X n ) The function L(θ) f(x| θ) is called the likelihood function Note: Terms that do not contain θ can be ignored in defining the likelihood

Example Write out the likelihood function for a random sample of size 10 from a (truncated exponential) distribution with pdf

Example Write out the likelihood function for a random sample of size n from a N(μ,σ 2 ) distribution

Example Write out the likelihood function for a random sample of size 10 from a Bernoulli distribution where 6 successes are observed, followed by 4 failures Write out the likelihood function for a random sample of size 10 from a Bernoulli distribution where 6 successes are observed How are these different?

Likelihood principle If different experiments based on a model defined by θ result in the same likelihood, one should draw the same conclusions

Properties of Sample Means and Proportions (8.7) Let X=(X 1, X 2,…,X n ) denote a random sample from some population The sample mean is: If This is a sample from a Bernoulli population, can denote this as:

Properties of Sample Means and Proportions When random sampling is performed, the sample observations are indentically distributed Thus, each X i come from a distribution with the same mean and same variance

Properties of Sample Means and Proportions If X=(X 1, X 2,…,X n ) represents random sample then the expected value of the sample mean is: The variance of the sample mean is:

Properties of Sample Means and Proportions If X=(X 1, X 2,…,X n ) represents random sample from a Bernoulli population, then the expected value of the sample proportion (sample mean) is: The variance of the sample proportion (sample mean) is:

Example A population of males has mean height of 70 inches and standard deviation of 3 inches For a random sample of size 4, what is the mean and variance of the sample mean For a random sample of size 40, what is the mean and variance of the sample mean

Properties of Sample Means from Normal Populations Suppose X=(X 1, X 2,…,X n ) represents random sample from a Normal population The distribution of the sample mean is:

Example A population of males has mean height of 70 inches and standard deviation of 3 inches For a random sample of size 4, what is the mean and variance of the sample mean For a random sample of size 40, what is the mean and variance of the sample mean