N t has 92 chapter 2 poisson processes independent

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: ding the joint density function of (T1 , T2 , T3 ) given that there were 3 arrivals before time t. The probability is 0 unless 0 < v1 < v2 < v3 < t. To compute the answer in this case, we note that P (N (t) = 4) = e t ( t)3 /3!, and in order to have T1 = t1 , T2 = t2 , T3 = t3 , N (t) = 4 we must have ⌧1 = t1 , ⌧2 = t2 t1 , ⌧3 = t3 t2 , and ⌧ > t t3 , so the desired conditional distribution is: e = = t1 (t2 t1 ) e e 3! =3 t ( t)3 /3! t 3 e ·e · e (t3 t)3 /3! t( t2 ) ·e (t t3 ) t Note that the answer does not depend on the values of v1 , v2 , v3 (as long as 0 < v1 < v2 < v3 < t), so the resulting conditional distribution is uniform over {(v1 , v2 , v3 ) : 0 < v1 < v2 < v3 < t} This set has volume t3 /3! since {(v1 , v2 , v3 ) : 0 < v1 , v2 , v3 < t} has volume t3 and v1 < v2 < v3 is one of 3! possible orderings. Generalizing from the concrete example it is easy to see that the joint density function of (T1 , T2 , . . . Tn ) given tha...
View Full Document

This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

Ask a homework question - tutors are online