1. Important facts

2. Review Reading pages: P330-P337 (6 th ), or P346-359 (7 th )

3. Chi-square distribution Definition If Xi are k independent, normally distributed random variables with mean 0 and variance 1, then the random variableindependentnormally distributedmeanvariance is distributed according to the chi-square distribution with k degrees of freedom. This is usually written The chi-square distribution has one parameter: k - a positive integer that specifies the number of degrees of freedom (i.e. the number of Xi)degrees of freedom The chi-square distribution is a special case of the gamma distribution. gamma distribution

4. Degree of Freedom Estimates of parameters can be based upon different amounts of information. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom (df).parameters In general, the degrees of freedom of an estimate is equal to the number of independent scores that go into the estimate minus the number of parameters estimated as intermediate steps in the estimation of the parameter itself. For example, if the variance, σ², is to be estimated from a random sample of N independent scores, then the degrees of freedom is equal to the number of independent scores (N) minus the number of parameters estimated as intermediate steps (one, μ estimated by sample mean) and is therefore equal to N-1.variance

5. Probability density function A probability density function of the chi-square distribution is probability density function where Γ denotes the Gamma function.Gamma function In mathematics, the Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. For a complex number z with positive real part the Gamma function is defined bymathematics GreekΓfactorialfunctionrealcomplex If n is a positive integer, then

8. t-distribution Student's t-distribution is the probability distribution of the ratio Where (i) Z is normally distributed with expected value 0 and variance 1;normally distributed expected value (ii) V has a chi-square distribution with ν degrees of freedom;chi-square distribution (iii) Z and V are independent.independent

9. Probability density function Student's t-distribution has the probability density function where ν is the number of degrees of freedom and Γ is the Gamma function.degrees of freedomGamma function

10. Density of the t-distribution (red and green) for 1, 2, 3, 5, 10, and 30 df compared to normal distribution (blue)

11. F-distribution A random variate of the F-distribution arises as the ratio of two chi-squared variates:random variatechi-squared where (i) U1 and U2 have chi-square distributions with d1 and d2 degrees of freedom respectively, and; (ii) U1 and U2 are independent.chi-square distributionsdegrees of freedomindependent

12. Probability density function The probability density function of an F(d1, d2) distributed random variable is given byprobability density functionrandom variable for real x ≥ 0, where d1 and d2 are positive integers, and B is the beta function.realpositive integersbeta function

13. Beta function In mathematics, the beta function is a special function defined bymathematics special function for x>0, and y>0.