density functionDensityFunction2002-09-11 21:59:152005-02-21 16:08:34Definition\usepackage{graphicx}
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\newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}}Let $X$ be a discrete random variable with sample space $\{x_1,x_2,\ldots\}$.
Let $p_k$ be the probability of $X$ taking the value $x_k$.
The function
$$
f(x)=\
\begin{cases}
p_k & \text{if }x=x_k\\
0 & \text{otherwise}
\end{cases}
$$
is called the \emph{probability function} or \emph{density function}.
It must hold:
$$\sum_{j=1}^{\infty} f(x_j)=1$$
If the density function for a random variable is known, we can calculate the probability of $X$ being on certain interval:
$$P[a<X\leq b] = \sum_{a<x_j\leq b}f(x_j) = \sum_{a<x_j\leq b}p_j.$$
The definition can be extended to continuous random variables in a direct way: The probability of $x$ being on a given interval is calculated with an integral instead of using a summation:
$$P[a<X\leq b] = \int_a^b f(x) dx.$$
For a more formal approach using measure theory, look at probability distribution function entry.