Continuous probability distribution

There are two main classes of probability distributions: Discrete and continuous. Discrete distributions describe variables that take on discrete values only (typically the positive integers), while continuous distributions describe variables that can take on arbitrary values in a continuum (typically the real numbers).

Introduction to continuous distributions

We may have a set of mutually exclusive propositions or possible outcomes that is neither discrete nor finite.

For instance, consider the following infinite set of propositions:

• The height of the next individual we'll meet is 0.0 meters.
• ...
• The height of the next individual we'll meet is 1.0 meters.
• ...

The height of an individual is a real number that in principle may take on any value.

The difficulty here is that there are too many propositions, we would never expect any one picked in advance to turn out to be the right one.

The solution is essentially a trick: One accepts that the probability for any one proposition is essentially zero (or infinitesimal), and instead concentrates on partitions of the set of propositions, in practice intervals in R. The probability assigned to an interval around f.i. 1.7 meters should thus be much larger than the one assigned to a similar size interval around f.i. 2.5 meters.

Formal definition

A continuous probability distribution is a function that, whren integrated over a set representing an event, gives the probability of that event. In other words,

$p(A)=\int _{A}f\,d\mu$.

In practice, the "event" A corresponds to a range of values, say $a\leq x\leq b$, so the above integral will tae the more familiar form

$p(a\leq x\leq b)=\int _{a}^{b}f\,dx$

Important examples

Gaussian distribution - Also known as the normal distribution.

Uniform continuous distribution -

Exponential distribution - Given a sequence of events, and the waiting time between two consequitive events is independent of how long we've already waited, the time between events follows the exponential distribution.

Gamma distribution -

Rayleigh distribution -

Cauchy distribution -

Laplacian distribution -

See also

• Entropy of a probability distribution

Related topics

• Stochastic variables
• Formal logic
• Measure theory
• Sigma algebra
• Quantum probability
• Stochastic convergence
• Stochastic diffential equations