Editing Likelihood function (section)

===Continuous probability distribution===
Let <math display="inline">X</math> be a [[random variable]] following an [[Probability distribution#Continuous probability distribution|absolutely continuous probability distribution]] with [[probability density function|density function]] <math display="inline">f</math> (a function of <math display="inline">x</math>) which depends on a parameter <math display="inline">\theta</math>. Then the function

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

considered as a function of <math display="inline">\theta</math>, is the ''likelihood function'' (of <math display="inline">\theta</math>, given the [[Outcome (probability)|outcome]] <math display="inline">X=x</math>). Again, <math display="inline">\mathcal{L}</math> is not a probability density or mass function over <math display="inline">\theta</math>, despite being a function of <math display="inline">\theta</math> given the observation <math display="inline">X = x</math>.

====Relationship between the likelihood and probability density functions====
The use of the [[probability density function|probability density]] in specifying the likelihood function above is justified as follows. Given an observation <math display="inline">x_j</math>, the likelihood for the interval <math display="inline">[x_j, x_j + h]</math>, where <math display="inline">h > 0</math> is a constant, is given by <math display="inline"> \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) </math>. Observe that
<math display="block"> \mathop\operatorname{arg\,max}_\theta \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) = \mathop\operatorname{arg\,max}_\theta \frac{1}{h} \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) ,</math>
since <math display="inline"> h </math> is positive and constant. Because
<math display="block"> 
\mathop\operatorname{arg\,max}_\theta \frac 1 h \mathcal{L}(\theta\mid x \in [x_j, x_j + h])
= \mathop\operatorname{arg\,max}_\theta \frac 1 h \Pr(x_j \leq x \leq x_j + h \mid \theta)
 = \mathop\operatorname{arg\,max}_\theta \frac 1 h \int_{x_j}^{x_j+h} f(x\mid \theta) \,dx,
</math>

where <math display="inline"> f(x\mid \theta) </math> is the probability density function, it follows that

<math display="block"> \mathop\operatorname{arg\,max}_\theta \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) = \mathop\operatorname{arg\,max}_\theta \frac{1}{h} \int_{x_j}^{x_j+h} f(x\mid\theta) \,dx .</math>

The first [[fundamental theorem of calculus]] provides that
<math display="block">
\lim_{h \to 0^{+}} \frac 1 h \int_{x_j}^{x_j+h} f(x\mid\theta) \,dx = f(x_j \mid \theta).
</math>

Then
<math display="block">
\begin{align}
\mathop\operatorname{arg\,max}_\theta \mathcal{L}(\theta\mid x_j)
&= \mathop\operatorname{arg\,max}_\theta \left[ \lim_{h\to 0^{+}} \mathcal{L}(\theta\mid x \in [x_j, x_j + h]) \right] \\[4pt]
&= \mathop\operatorname{arg\,max}_\theta \left[ \lim_{h\to 0^{+}} \frac{1}{h} \int_{x_j}^{x_j+h} f(x\mid\theta) \,dx \right] \\[4pt]
&= \mathop\operatorname{arg\,max}_\theta f(x_j \mid \theta).
\end{align}
</math>

Therefore,
<math display="block"> \mathop\operatorname{arg\,max}_\theta \mathcal{L}(\theta\mid x_j) = \mathop\operatorname{arg\,max}_\theta f(x_j \mid \theta), </math>
and so maximizing the probability density at <math display="inline"> x_j </math> amounts to maximizing the likelihood of the specific observation <math display="inline"> x_j </math>.