STAT_333_Assignment_2_Solutions

# STAT_333_Assignment_2_Solutions - STAT 333 Assignment 2...

This preview shows pages 1–2. Sign up to view the full content.

STAT 333 Assignment 2 SOLUTIONS 1. Prove the Delayed Renewal Relation. (Hint: the proof is very similar to that of the Renewal Relation, and you have to be very careful with the starting values of sums.) For n ≥ 1, d n = P(λ occurs on trial n) = 𝑃 ( λ occurs first on trial k, and also on trial n) 𝑘 =1 = 𝑃 λ occurs first on trial k P( λ occurs on trial n|on k) 𝑘 =1 = f k P( λ occurs n k trials after λ ) 𝑘 =1 = f k r n k 𝑘 =1 = f k r n k 𝑘 =0 since f 0 = 0 And 𝐷 λ ? = 𝑑 𝑛 ? 𝑛 𝑛 =0 = 0 + 𝑑 𝑛 ? 𝑛 𝑛 =1 since d 0 = 0 = ( f k r n k 𝑘 =0 ) ? 𝑛 𝑛 =1 = ( f k r n k 𝑘 =0 ) ? 𝑛 𝑛 =0 since the n = 0 inner sum is 0 = ( 𝑓 𝑛 ? 𝑛 )( r n s n 𝑘 =0 ) 𝑛 =0 since 𝑑 𝑛 = 𝑓 𝑛 × ? 𝑛 Thus 𝐷 λ ? = 𝐹 λ ? 𝑅 λ ? . Rearranging gives the required result. 2. A fair 6- sided die is rolled repeatedly. Let λ be the event “the maximum roll so far is ≤ 5” a. Explain carefully why λ is a renewal event. The first waiting time is 1 if the first roll is 5 and ∞ if it is 6. The between-event waiting time has the exact same distribution. Given it has occurred once, the first roll was 5. For the maximum to continue to be ≤ 5 , the second waiting time is either 1 (if the second roll is ≤ 5) or ∞ if it is 6. All waiting times have the distribution: 1 with prob 5/6 ∞ with prob 1/6 And they are independent. Thus λ is a renewal event. b. Determine the renewal sequence { r n } r 0 = 1 r 1 = P(first roll is ≤ 5) = 5/6 r 2 = P(min is ≤ 5) = P(first and second rolls are ≤ 5) = (5/6) 2 since rolls are independent r n = (5/6) n for n ≥ 1 c. Find f λ by any legitimate method. Is λ recurrent or transient?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern