
related topics 
{math, number, function} 
{rate, high, increase} 
{math, energy, light} 
{household, population, female} 

In probability theory and statistics, a probability distribution identifies either the probability of each value of a random variable (when the variable is discrete), or the probability of the value falling within a particular interval (when the variable is continuous).^{[1]} The probability distribution describes the range of possible values that a random variable can attain and the probability that the value of the random variable is within any (measurable) subset of that range.
When the random variable takes values in the set of real numbers, the probability distribution is completely described by the cumulative distribution function, whose value at each real x is the probability that the random variable is smaller than or equal to x.
The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics many processes are described probabilistically, from the kinetic properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.
There are various probability distributions that show up in various different applications. Two of the most important ones are the normal distribution and the categorical distribution. The normal distribution, also known as the Gaussian distribution, has a familiar "bell curve" shape and approximates many different naturally occurring distributions over real numbers. The categorical distribution describes the result of an experiment with a fixed, finite number of outcomes. For example, the toss of a fair coin is a categorical distribution, where the possible outcomes are heads and tails, each with probability 1/2.
Contents
Full article ▸


related documents 
Autocorrelation 
Mersenne twister 
Controllability 
Lebesgue measure 
Linear 
Probability space 
Golomb coding 
Symmetric group 
Search algorithm 
Euler's totient function 
Division algebra 
Constant of integration 
Mersenne prime 
Exact sequence 
Separable space 
Kruskal's algorithm 
Analytic geometry 
Convex set 
Carmichael number 
Local ring 
Linear equation 
Inverse limit 
Banach fixed point theorem 
Goodstein's theorem 
Group representation 
Automated theorem proving 
Wellorder 
Diophantine set 
Cardinality 
Kolmogorov space 
