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Let us now apply what we have just learned about the mean, variance, and
standard deviation of a general probability distribution function
to the specific case of the
binomial probability distribution. Recall, from Section 2.6,
that if a simple system has just two possible outcomes,
denoted 1 and 2, with
respective probabilities
and
,
then the probability of obtaining
occurrences of outcome 1 in
observations is
![$\displaystyle P_N(n_1) = \frac{N!}{n_1 ! (N-n_1)!} p^{ n_1} q^{ N-n_1}.$](img137.png) |
(2.38) |
Thus, making use of Equation (2.27), the mean number of occurrences of outcome 1 in
observations
is given by
![$\displaystyle \overline{n_1} = \sum_{n_1=0,N} P_N(n_1) n_1 = \sum_{n_1=0,N} \frac{N!}{n_1! (N-n_1)!} p^{ n_1} q^{ N-n_1} n_1.$](img184.png) |
(2.39) |
We can see that if the
final factor
were absent on the right-hand side of the previous expression then it would just reduce to the binomial expansion, which we
know how to sum. [See Equation (2.23).] We can take advantage of this fact using a rather elegant
mathematical sleight of hand. Observe that because
![$\displaystyle n_1 p^{ n_1} \equiv p \frac{\partial}{\partial p} p^{ n_1},$](img185.png) |
(2.40) |
the previous summation can be rewritten as
![$\displaystyle \sum_{n_1=0,N}\frac{N!}{n_1! (N-n_1)!} p^{ n_1} q^{ N-n_1} ...
...\left[\sum_{n_1=0,N} \frac{N!}{n_1! (N-n_1)!} p^{ n_1} q^{ N-n_1} \right].$](img186.png) |
(2.41) |
The term in square brackets is now the familiar binomial expansion, and
can be written more succinctly as
.
Thus,
![$\displaystyle \sum_{n_1=0,N}\frac{N!}{n_1! (N-n_1)!} p^{ n_1} q^{ N-n_1} n_1 =p \frac{\partial}{\partial p} (p+q)^{ N}= p N (p+q)^{ N-1}.$](img187.png) |
(2.42) |
However,
for the case in hand [see Equation (2.11)], so
![$\displaystyle \overline{n_1} = N p.$](img188.png) |
(2.43) |
In fact, we could have guessed the previous result.
By definition, the probability,
, is the number of occurrences of the
outcome 1 divided by the number of trials, in the limit as the number
of trials goes to infinity:
![$\displaystyle p= _{lt N\rightarrow\infty }\frac{n_1}{N}.$](img189.png) |
(2.44) |
If we think carefully, however,
we can see that taking the limit as the number
of trials goes to infinity is equivalent to taking the mean value,
so that
![$\displaystyle p = \overline{\left(\frac{n_1}{N}\right)} = \frac{\overline{n_1}}{N}.$](img190.png) |
(2.45) |
But, this is just a simple rearrangement of Equation (2.43).
Let us now calculate the variance of
. Recall, from Equation (2.36), that
![$\displaystyle \overline{({\mit\Delta} n_1)^{ 2}}= \overline{(n_1)^{ 2}} - (\overline{n_1})^{ 2}.$](img191.png) |
(2.46) |
We already know
,
so we just need to calculate
.
This average is written
![$\displaystyle \overline{(n_1)^{ 2}}=\sum_{n_1=0,N}\frac{N!}{n_1! (N-n_1)!} p^{ n_1} q^{ N-n_1} (n_1)^{ 2}.$](img194.png) |
(2.47) |
The sum can be evaluated using a simple extension of the mathematical trick that
we used previously to evaluate
. Because
![$\displaystyle (n_1)^{ 2} p^{ n_1} \equiv \left(p \frac{\partial}{\partial p}\right)^{ 2} p^{ n_1},$](img195.png) |
(2.48) |
then
Using
, we obtain
because
. [See Equation (2.43).] It follows that the variance
of
is given by
![$\displaystyle \overline{({\mit\Delta} n_1)^{ 2}}= \overline{(n_1)^{ 2}}- \left(\overline{n_1}\right)^{ 2} = N p q.$](img205.png) |
(2.51) |
The standard deviation of
is the square root of the variance [see Equation (2.37)], so that
![$\displaystyle {\mit\Delta}^\ast n_1 = \sqrt{N p q}.$](img206.png) |
(2.52) |
Recall that this quantity is essentially the width of the range over which
is distributed around its mean value. The relative width of the
distribution is characterized by
![$\displaystyle \frac{{\mit\Delta}^\ast n_1}{\overline{n_1}}= \frac{\sqrt{N p q}}{N p} = \sqrt{\frac{q}{p}}\frac{1}{\sqrt{N}}.$](img207.png) |
(2.53) |
It is clear, from this formula, that the relative width decreases with increasing
like
. So, the greater the number of trials, the
more likely it is that an observation of
will yield a result
that is relatively close to the mean value,
.
Next: Gaussian Probability Distribution
Up: Probability Theory
Previous: Mean, Variance, and Standard
Richard Fitzpatrick
2016-01-25