Math/Stat 511 R. Sharpley

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

2. 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. 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. Definition of the Cumulative Distribution

5. Complementary event

6. Definition of the random variable W

7. The Poisson distribution of the sum of ‘changes’ k is less or equal ; here the parameter for the Poisson is (w).

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

9. If we pull off the first term of the series, this becomes

10. If we differentiate this last expression, i.e. we obtain

11. To verify these steps, observe Differentiate the sum, ...

12. To verify these steps, observe Differentiate the sum, applying the product rule.

13. To verify these steps, observe Notice the sum telescopes to give

14. To verify these steps, observe which algebraically reduces to

15. 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.