Mathematical Methods for Physicists: A concise introduction - Site Map

INTRODUCTION TO PROBABILITY THEORY
Why is the discrete probability function (14.16) called the binomial distribution?
Since for m ˆ 0; 1; 2; ...; n it corresponds to successive terms in the binomial
expansion
…q ‡ p† n ˆ q n ‡ n C 1 q n1 p ‡ n C 2 q n2 p 2 ‡‡p n ˆ Xn
mˆ0
nC m p m q nm :
To illustrate the use of the binomial distribution (14.16), let us ®nd the probability
that a one will appear exactly 4 times if a die is thrown 10 times. Here
n ˆ 10, m ˆ 4, p ˆ 1=6, and q ˆ…1 p† ˆ5=6. Hence the probability is
f …4† ˆP…X ˆ 4† ˆ 10!
1 4
5 6ˆ 0:0543:
4!6! 6 6
A few examples of binomial distributions, computed from Eq. (14.16), are
shown in Figs. 14.2, and 14.3 by means of histograms.
One of the key requirements for a probability distribution is that
X n
mˆo
f …m† ˆXn
mˆo
nC m p m q nm ˆ 1:
…14:17†
To show that this is in fact the case, we note that
X n
mˆo
nC m p m q nm
Figure 14.2. The distribution is symmetric about m ˆ 10:
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