t distribution

1. t distribution • Suppose Z ~ N(0,1) independent of X ~ χ2(n). Then, that is, T has a t distribution with n degrees of freedom. • The probability density function of T is given by • Properties of the t distribution … week3

2. Important Use of the t Distribution • Suppose X1, X2,…Xn are i.i.d normal random variables with mean μ and variance σ2. Then, • Proof: week3

3. Cauchy Distribution • Let Z1, Z2 be independent N(0,1) variables. Then…. week3

4. F distribution • Suppose X ~ χ2(n) independent of Y ~ χ2(m). Then, • Important use of this is …. week3

5. Properties of the F distribution • The F-distribution is a right skewed distribution. • i.e. • Can use Table A.9 on page 794 to find percentile of the F- distribution. • Example… week3

6. Claim • The square of random variable with t(n) distribution has an F distribution with (1, n) df. That is, • Proof: week3

7. Parameters and Point Estimate • Distributions have parameters. Parameters are usually denoted using the letter θ. • A point estimate, is a single number, usually calculated from the sample data, that we use to estimate an unknown parameter θ. • A point estimator is a statistic (i.e. a function) that tells us how we can use the sample data to create a numeric point estimate. • A point estimator and a point estimate is usually denoted by • Examples:… week3

8. Assessing an Estimator • For any parameter, there are many different point estimators of θ. How do we know which point estimators are “good”? • There are few criteria… • The bias of an estimator is We would like our estimator to have zero (or very small) bias. • The variance of an estimator is We would like our estimator to have small variance. • Small bias and small variance are usually competing goals, often we can’t minimize both properties. week3

9. Mean Square Error of Point Estimators • The mean square error (MSE) of a point estimator is • The Mean Square Error of an estimator combines bias and variance, we want our estimator to have small MSE. • Claim: week3

10. Example week3

11. Minimum Variance Unbiased Estimator • MVUE is the unbiased estimator with the smallest possible variance. Look amongst all unbiased estimators for the one with the smallest variance. • Note, is called the Standard Error of a point estimator. week3