# N t has 92 chapter 2 poisson processes independent

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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 &lt; v1 &lt; v2 &lt; v3 &lt; 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 ⌧ &gt; 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 &lt; v1 &lt; v2 &lt; v3 &lt; t), so the resulting conditional distribution is uniform over {(v1 , v2 , v3 ) : 0 &lt; v1 &lt; v2 &lt; v3 &lt; t} This set has volume t3 /3! since {(v1 , v2 , v3 ) : 0 &lt; v1 , v2 , v3 &lt; t} has volume t3 and v1 &lt; v2 &lt; 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).

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