1. 511 Friday March 30, 2001 Math/Stat 511 R. Sharpley Lecture #27: a. Verification of the derivation of the gamma random variable b.Begin the standard normal random variable

2. 511 Friday March 30, 2001 We wish to fill in the reasoning to verify the steps in the derivation of the gamma model on page 186 of the Text. The formula for the gamma probability density function is To verify this we go to equation 4.3-1 on page 186, which reads:

3. 511 Friday March 30, 2001 This is to verify the steps in the derivation of the gamma model on page 186 of the Text. Equation 4.3-1 reads

4. 511 Friday March 30, 2001 Definition of the Cumulative Distribution

5. 511 Friday March 30, 2001 Complementary event

6. 511 Friday March 30, 2001 Definition of the random variable W

7. 511 Friday March 30, 2001 The Poisson distribution of the sum of ‘changes’ k is less or equal  ; here the parameter for the Poisson is ( w).

8. 511 Friday March 30, 2001 This is the probability that there are k changes in [0,w], i.e. with parameter of expected changes now equal to ( w).

9. 511 Friday March 30, 2001 If we pull off the first term of the series, this becomes

10. 511 Friday March 30, 2001 If we differentiate this last expression, i.e. we obtain

11. 511 Friday March 30, 2001 Differentiate the sum,... To verify these steps, observe

12. 511 Friday March 30, 2001 Differentiate the sum, applying the product rule. To verify these steps, observe

13. 511 Friday March 30, 2001 Notice the sum telescopes to give To verify these steps, observe

14. 511 Friday March 30, 2001 which algebraically reduces to To verify these steps, observe

15. 511 Friday March 30, 2001 So if  := 1/, then This is the pdf random variable which models the waiting time for at least  ‘changes’ of a Poisson process with  equal to the mean waiting time for the first change.