Editing Maximum likelihood estimation (section)

=== Discrete distribution, continuous parameter space ===
Now suppose that there was only one coin but its {{mvar|p}} could have been any value {{nowrap| 0 ≤ {{mvar|p}} ≤ 1 .}} The likelihood function to be maximised is
<math display="block">
L(p) = f_D(\mathrm{H} = 49 \mid p) = \binom{80}{49} p^{49}(1 - p)^{31}~,
</math>

and the maximisation is over all possible values {{nowrap|0 ≤ {{mvar|p}} ≤ 1 .}}

[[File:MLfunctionbinomial-en.svg|thumb|200px|Likelihood function for proportion value of a binomial process ({{mvar|n}}&nbsp;=&nbsp;10)]]

One way to maximize this function is by [[derivative|differentiating]] with respect to {{mvar|p}} and setting to zero:

<math display="block">\begin{align}
0 & = \frac{\partial}{\partial p} \left( \binom{80}{49} p^{49}(1-p)^{31} \right)~, \\[8pt]
0 & = 49 p^{48}(1-p)^{31} - 31 p^{49}(1-p)^{30} \\[8pt]
  & = p^{48}(1-p)^{30}\left[ 49 (1-p) - 31 p \right]  \\[8pt]
  & = p^{48}(1-p)^{30}\left[ 49 - 80 p \right]~.
\end{align}</math>

This is a product of three terms.  The first term is 0 when {{mvar|p}}&nbsp;=&nbsp;0.  The second is 0 when {{mvar|p}}&nbsp;=&nbsp;1.  The third is zero when {{mvar|p}}&nbsp;=&nbsp;{{frac|49|80}}. The solution that maximizes the likelihood is clearly {{mvar|p}}&nbsp;=&nbsp;{{frac|49|80}} (since {{mvar|p}}&nbsp;=&nbsp;0 and {{mvar|p}}&nbsp;=&nbsp;1 result in a likelihood of 0). Thus the ''maximum likelihood estimator'' for {{mvar|p}} is {{frac|49|80}}.

This result is easily generalized by substituting a letter such as {{mvar|s}} in the place of 49 to represent the observed number of 'successes' of our [[Bernoulli trial]]s, and a letter such as {{mvar|n}} in the place of 80 to represent the number of Bernoulli trials. Exactly the same calculation yields {{frac|{{mvar|s}}|{{mvar|n}}}} which is the maximum likelihood estimator for any sequence of {{mvar|n}} Bernoulli trials resulting in {{mvar|s}} 'successes'.