Editing Probability theory (section)

===Continuous probability distributions===
{{Main|Continuous probability distribution}}

[[File:Gaussian distribution 2.jpg|thumb|300px|The [[normal distribution]], a continuous probability distribution]]

{{em|Continuous probability theory}} deals with events that occur in a continuous sample space.

{{em|Classical definition}}:
The classical definition breaks down when confronted with the continuous case. See [[Bertrand's paradox (probability)|Bertrand's paradox]].

{{em|Modern definition}}:
If the sample space of a random variable ''X'' is the set of [[real numbers]] (<math>\mathbb{R}</math>) or a subset thereof, then a function called the {{em|[[cumulative distribution function]]}} ({{em|CDF}}) <math>F\,</math> exists, defined by <math>F(x) = P(X\le x) \,</math>. That is, ''F''(''x'') returns the probability that ''X'' will be less than or equal to ''x''.

The CDF necessarily satisfies the following properties.
# <math>F\,</math> is a [[Monotonic function|monotonically non-decreasing]], [[right-continuous]] function;
# <math>\lim_{x\rightarrow -\infty} F(x)=0\,;</math>
# <math>\lim_{x\rightarrow \infty} F(x)=1\,.</math>

The random variable <math>X</math> is said to have a continuous probability distribution if the corresponding CDF <math>F</math> is continuous. If <math>F\,</math> is [[absolutely continuous]], then its derivative exists almost everywhere and integrating the derivative gives us the CDF back again. In this case, the random variable ''X'' is said to have a {{em|[[probability density function]]}} ({{em|PDF}}) or simply {{em|density}} <math>f(x)=\frac{dF(x)}{dx}\,.</math>

For a set <math>E \subseteq \mathbb{R}</math>, the probability of the random variable ''X'' being in <math>E\,</math> is
:<math>P(X\in E) = \int_{x\in E} dF(x)\,.</math>

In case the PDF exists, this can be written as
:<math>P(X\in E) = \int_{x\in E} f(x)\,dx\,.</math>

Whereas the ''PDF'' exists only for continuous random variables, the ''CDF'' exists for all random variables (including discrete random variables) that take values in <math>\mathbb{R}\,.</math>

These concepts can be generalized for [[Dimension|multidimensional]] cases on <math>\mathbb{R}^n</math> and other continuous sample spaces.