Mathematical Statistics—Important Distributions based on Normal Random Variables

A Few Useful Results from Probability Theory

· The chi-squared distribution (with degrees of freedom) is a special case of the gamma distribution :

o Recall the Devore & Berk textbook uses a parameterization of the gamma distribution that is different from the Ross textbook. For a review of the gamma distribution, see Section 4.4 of the Devore & Berk text. Also, recall the gamma function is . (The gamma function has interesting properties—you can review these in Section 4.4.)

o The chi-squared distribution is positively skewed (longer right tail), but becomes more symmetric as , the degrees of freedom, increases.

o The chi-squared distribution is denoted .

o The mean of the distribution is and the variance is . Also the moment-generating function is .

o Probabilities (areas under the curve) for the chi-squared distribution are given in Table A.7 of the textbook.

· If are independent gamma random variables, then is a gamma random variable. (This is easy to show using the moment-generating-function technique. We showed the result last term in Probability Theory.)

· Applying the last result, if are independent chi-squared random variables with degrees of freedom , respectively, then is a chi-squared random variable with degrees of freedom.

· If are independent standard normal random variables (i.e., mean = 0, standard deviation = 1), then is a chi-squared random variable with n degrees of freedom. (We showed this result, using the distribution-function technique, in Probability Theory.)