Editing Likelihood function (section)

===Likelihoods for mixed continuous&ndash;discrete distributions===
The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete probability masses <math display="inline">p_k (\theta)</math> and a density <math display="inline">f(x\mid\theta)</math>, where the sum of all the <math display="inline">p</math>'s added to the integral of <math display="inline">f</math> is always one. Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with in the manner shown above. For an observation from the discrete component, the likelihood function for an observation from the discrete component is simply
<math display="block">\mathcal{L}(\theta \mid x )= p_k(\theta), </math>
where <math display="inline">k</math> is the index of the discrete probability mass corresponding to observation <math display="inline">x</math>, because maximizing the probability mass (or probability) at <math display="inline">x</math> amounts to maximizing the likelihood of the specific observation.

The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and the probability mass) arises from the way in which the likelihood function is defined up to a constant of proportionality, where this "constant" can change with the observation <math display="inline">x</math>, but not with the parameter <math display="inline">\theta</math>.