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Unformatted text preview: l go between 12:31 and 12:33 is
j !k !(n j k )! ✓ ◆j ✓
n ◆k ✓ 1 3
n ◆n (j +k) Rearranging gives
( )j (2 )k n(n
k! 1) · · · (n j
nj +k k + 1) ✓ ·1 3
n ◆n (j +k) Reasoning as before shows that when n is large, this is approximately
( )j (2 )k
Writing e =e /3 e 2 /3 3 and rearranging we can write the last expression as
j e j! ·e 2 (2 )k
k! This shows that the number of arrivals in the two time intervals we chose are
independent Poissons with means and 2 .
The last proof can be easily generalized to show that if we divide the hour
between 12:00 and 1:00 into any number of intervals, then the arrivals are
independent Poissons with the right means. However, the argument gets very
messy to write down. More realistic models.
Two of the weaknesses of the derivation above are:
(i) All students are assumed to have exactly the same probability of going to
the Great Hall.
(ii) The probability of going in a given time interval is a constant multiple of
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
- Spring '10
- The Land