Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL PROBABILITY DISTRIBUTIONS
and the standard deviation ;
ˆ
p npq : …14:24†
Two di€erent limits of the binomial distribution for large n are of practical
importance: (1) n !1and p ! 0 in such a way that the product np ˆ remains
constant; (2) both n and pn are large. The ®rst case will result a new distribution,
the Poisson distribution, and the second cases gives us the Gaussian (or Laplace)
distribution.
The Poisson distribution
Now np ˆ ; so p ˆ =n. The binomial distribution (14.16) then becomes
n! m
f …m† ˆP…X ˆ m† ˆ
1 nm
m!…n m†! n n
n…n 1†…n 2†…n m ‡ 1†
ˆ
m!n m m 1 nm
n
ˆ 1 1
1 2
1 m 1
m
1 nm
: …14:25†
n n
n m! n
Now as n !1,
1 1 n
1 2
n
1 m 1
! 1;
n
while
1 nmˆ 1 n
1 m
! e
… 1†
ˆ e ;
n
n n
where we have made use of the result
lim 1 ‡ nˆ e :
n!1 n
It follows that Eq. (14.25) becomes
f …m† ˆP…X ˆ m† ˆm e
: …14:26†
m!
This is known as the Poisson distribution. Note that P 1
mˆ0
P…X ˆ m† ˆ1; as it
should.
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