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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...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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