1. Statistics 270 - Lecture 16

2. Gamma Distribution Normal pdf is symmetric and bell-shaped Not all distributions have these properties Some pdf’s give a skewed distribution One such family is the gamma family

3. Gamma Function For  >0, the gamma function is Properties: For any  >1,  (  )=(  -1)  (  -1) For any positive integer, n,  (n)=(n-1)!  ( 1/2)=

4. Gamma PDF A continuous random variable, X, has a gamma distribution if it has pdf: Where,  >0 and  >0

5. Gamma PDF Expected Value: Variance:

6. Standard Gamma PDF’s Standard gamma=incomplete gamma  = 1 Table A.4 gives tabled probabilities for incomplete gamma pdf’s for some  ’s

7. Example: (Chapter 4, #57) Suppose the time spent by a randomly selected student who uses a terminal connected to a local computer facility has a mean of 20 minutes and variance 80 minutes 2 Find  and  ? What is the probability that the student uses the terminal less than 24 minutes?

8. Example: (Chapter 4, #57) Suppose that X has a gamma(4.5,1.5) distribution Find

9. Comments

10. Exponential Distribution

11. Memoryless Property