4.2.4 An example: Fitting  4.2 Applications  4.2.2 Mean of the

4.2.3 Mean of the binomial probability distribution function

The binomial distribution function describes the probability of observing n events in a given class out of N trials, when the population-average probability for the given class of event is p:

$$\Phi_B(n,p) = \left({{N}\atop{n}}\right) p^n (1-p)^{N-n} ~.$$ (4.18)

 

If a trial is conducted and n^* events are observed, what is the best estimate for the parameter p? The logarithm of the likelihood function for p is

$$W(p) = n {\rm ln} p + (N-n){\rm ln} (1-p) + {\rm constant} ~.$$ (4.19)

 

The maximum likelihood occurs for

$${{\partial W}\over{\partial p}} = 0 = {{n}\over{p}} - {{(N-n)}\over{(1-p)}} = {{(n-pN)}\over{p(1-p)}} \ .$$ (4.20)

 

Then

$$p^* = {{n^*}\over{N}} ~.$$ (4.21)

 

is the resulting maximum-likelihood estimator for p.

The uncertainty in p can be found by use of

$${{\partial^2W}\over{\partial p^2}} = - {{N}\over{p^* (1-p^*)}}$$ (4.22)

 

in (4.11):

$$\sigma_{p^*} = \Bigl[{{p^*(1-p^*)}\over{N}}\Bigr]^{1/2} \ .$$ (4.23)

 

Note that the standard deviation in n,

$$\sigma_n = N \sigma_p = \bigl(Np^*(1-p^*)\bigr)^{1/2} \ ,$$ (4.24)

 

is smaller than $\sqrt{N}.$

   
 4.2.4 An example: Fitting  4.2 Applications  4.2.2 Mean of the