The (probability) density

A random variable X is called continuous, if there exists an integrable function \(f: R -> R\) with \(f(x) \ge 0\) for all \(x in R\) such that for every interval [a,b]:

\[P(a \le X \le b) = \int_{a}^{b} f(x) \; dx\]

The function f is called the (probability) density of X. The set \(T = {x \in R: f(x) > 0}\) is called the support of X resp. f.

The density is the derivation of the distribution function. So if you want to calculate out of the density you have to build the indefinite integral of f(x)

To be a density function, f(x) has to fulfill the following term: \(\int_{a}^{b} f(x) \; dx = 1\)

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For continuous uniform distribution:

\[f(x) = \begin{cases} \frac{1}{b - a} ,& a \le x \le b\\ 0 ,& otherwise \end{cases} \]

For the exponential distribution

\[f(x) = \begin{cases} \lambda e^{-\lambda x} ,& x \ge 0\\ 0 ,& x < 0 \end{cases} \]

For the normal distribution

\[f(x) = \frac{1}{\sqrt(2 \pi o)} exp(- \frac{(x - \mu)^2}{2o^2})\]

specifically for \(\mu = 0\) and \(o^2 = 1\), the density \(\phi\) is: \(\frac{1}{\sqrt(2\pi)} exp(-0.5x^2)\)

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