Today Today: More on the Normal Distribution (section 6.1), begin Chapter 8 (8.1 and 8.2) 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: –Section 6.1 –Sections 8.1, 8.2, 8.4, 8.7, 8.8, 8.10

Example: Suppose Z~N(0,1) Find P(Z<2.00) Find P(Z<1.96) Find P(Z>3.4) Find P(Z<6) Find P(1.96<Z<2)

Example: The height of female students at a University follows a normal distribution with mean of 65 inches and standard deviation of 2 inches Find the probability that a randomly selected female student has a height less than 58 inches What is the 99 th percentile of this distribution?

Finding a Percentile Can use the relationship between Z and the random variable X to compute percentiles for the distribution of X The 100p th percentile of normally distributed random variable X with mean μ and variance σ can be found using the standard normal distribution

Example: The height of female students at a University follows a normal distribution with mean of 65 inches and standard deviation of 2 inches What is the 99 th percentile of this distribution?

Properties of the Normal Distribution Is a bell shaped distribution, centered at μ Is a symmetric distribution If X~N(μ,σ 2 ), then (a+bX)~ N(a+bμ,b 2 σ 2 ) If X 1, X 2,…,X n are independent random variables, such that X i ~N(μ i,σ i 2 ), then Σ a i X i ~N( Σ a i μ i, Σ a i 2 σ i 2 )

Example Let X i ~N(5,4), for i=1,2,…n What is the distribution of Y=4X 1 +6 Find P(Y<20) What is the distribution of Σ x i ? Write out its pdf

Chapter 8 Samples and Statistics We have considered several probability models (e.g., N(0,1)) Often, believe a system follows a particular model, but do not know the values of the parameters of the model

Some Definitions To estimate a population characteristic, we use statistics Can view a statistic as a function of data The data come from a sample collected from the population of interest The characteristic of interest is called the population parameter which is estimated by a sample statistic

Some Definitions A statistic T=g(X 1, X 2,…,X n ), a function of the sample observations X 1, X 2,…,X n mat vary from sample to sample Thus a statistic T has a probability distribution of its own called the sampling distribution of the statistic

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:

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