Editing Likelihood function (section)

===Discrete probability distribution===
Let <math display="inline">X</math> be a discrete [[random variable]] with [[probability mass function]] <math display="inline">p</math> depending on a parameter <math display="inline">\theta</math>. Then the function

<math display="block">\mathcal{L}(\theta \mid x) = p_\theta (x) = P_\theta (X=x), </math>

considered as a function of <math display="inline">\theta</math>, is the ''likelihood function'', given the [[Outcome (probability)|outcome]] <math display="inline">x</math> of the random variable <math display="inline">X</math>. Sometimes the probability of "the value <math display="inline">x</math> of <math display="inline">X</math> for the parameter value <math display="inline">\theta</math>{{resize|20%|&nbsp;}}" is written as {{math|''P''(''X'' {{=}} ''x'' {{!}} ''θ'')}} or {{math|''P''(''X'' {{=}} ''x''; ''θ'')}}. The likelihood is the probability that a particular outcome <math display="inline">x</math> is observed when the true value of the parameter is <math display="inline">\theta</math>, equivalent to the probability mass on <math display="inline">x</math>; it is ''not'' a probability density over the parameter <math display="inline">\theta</math>. The likelihood, <math display="inline">\mathcal{L}(\theta \mid x) </math>, should not be confused with <math display="inline">P(\theta \mid x)</math>, which is the posterior probability of <math display="inline">\theta</math> given the data <math display="inline">x</math>.

====Example====
[[Image:likelihoodFunctionAfterHH.png|thumb|400px|Figure 1.&nbsp; The likelihood function (<math display="inline">p_\text{H}^2</math>) for the probability of a coin landing heads-up (without prior knowledge of the coin's fairness), given that we have observed HH.]]
[[Image:likelihoodFunctionAfterHHT.png|thumb|400px|Figure 2.&nbsp; The likelihood function (<math display="inline">p_\text{H}^2(1-p_\text{H})</math>) for the probability of a coin landing heads-up (without prior knowledge of the coin's fairness), given that we have observed HHT.]]
Consider a simple statistical model of a coin flip: a single parameter <math display="inline">p_\text{H}</math> that expresses the "fairness" of the coin. The parameter is the probability that a coin lands heads up ("H") when tossed. <math display="inline">p_\text{H}</math> can take on any value within the range 0.0 to 1.0. For a perfectly [[fair coin]], <math display="inline">p_\text{H} = 0.5</math>.

Imagine flipping a fair coin twice, and observing two heads in two tosses ("HH"). Assuming that each successive coin flip is [[Independent and identically distributed random variables|i.i.d.]], then the probability of observing HH is

<math display="block">P(\text{HH} \mid p_\text{H}=0.5) = 0.5^2 = 0.25.</math>

Equivalently, the likelihood of observing "HH" assuming <math display="inline">p_\text{H} = 0.5</math> is

<math display="block">\mathcal{L}(p_\text{H}=0.5 \mid \text{HH}) = 0.25.</math>

This is not the same as saying that <math display="inline">P(p_\text{H} = 0.5 \mid HH) = 0.25</math>, a conclusion which could only be reached via [[Bayes' theorem]] given knowledge about the marginal probabilities  <math display="inline">P(p_\text{H} = 0.5)</math> and <math display="inline">P(\text{HH})</math>.

Now suppose that the coin is not a fair coin, but instead that <math display="inline">p_\text{H} = 0.3</math>. Then the probability of two heads on two flips is

<math display="block">P(\text{HH} \mid p_\text{H}=0.3) = 0.3^2 = 0.09.</math>

Hence

<math display="block">\mathcal{L}(p_\text{H}=0.3 \mid \text{HH}) = 0.09.</math>

More generally, for each value of <math display="inline">p_\text{H}</math>, we can calculate the corresponding likelihood. The result of such calculations is displayed in Figure&nbsp;1. The integral of <math display="inline">\mathcal{L}</math> over [0,&nbsp;1] is 1/3; likelihoods need not integrate or sum to one over the parameter space.