Editing Random variable (section)

===Continuous random variable===

Formally, a continuous random variable is a random variable whose [[cumulative distribution function]] is [[Continuous function|continuous]] everywhere.<ref name=":0">{{Cite book|title=Introduction to Probability|last=Bertsekas|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν.|isbn=188652940X|location=Belmont, Mass.|oclc=51441829}}</ref> There are no "[[Discontinuity (mathematics)#Jump discontinuity|gaps]]", which would correspond to numbers which have a finite probability of [[Outcome (probability)|occurring]].  Instead, continuous random variables [[almost never]] take an exact prescribed value ''c'' (formally, <math display="inline">\forall c \in \mathbb{R}:\; \Pr(X = c) = 0</math>) but there is a positive probability that its value will lie in particular [[Interval (mathematics)|intervals]] which can be [[arbitrarily small]].  Continuous random variables usually admit [[probability density function]]s (PDF), which characterize their CDF and [[probability measure]]s; 
such distributions are also called [[Absolutely continuous random variable|absolutely continuous]]; but some continuous distributions are [[Singular distribution|singular]], or mixes of an absolutely continuous part and a singular part.

An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, '''''X''''' = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any ''range'' of values. For example, the probability of choosing a number in [0, 180] is {{frac|1|2}}. Instead of speaking of a probability mass function, we say that the probability ''density'' of '''''X''''' is 1/360. The probability of a subset of [0,&nbsp;360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.

More formally, given any [[Interval (mathematics)|interval]] <math display="inline">I = [a, b] = \{x \in \mathbb{R} : a \le x \le b \}</math>, a random variable <math>X_I \sim \operatorname{U}(I) = \operatorname{U}[a, b]</math> is called a "[[Continuous uniform distribution|continuous uniform]] random variable" (CURV) if the probability that it takes a value in a [[subinterval]] depends only on the length of the subinterval. This implies that the probability of <math>X_I</math> falling in any subinterval <math>[c, d] \sube [a, b]</math> is [[Proportionality (mathematics)|proportional]] to the [[Lebesgue measure|length]] of the subinterval, that is, if {{math|''a'' ≤ ''c'' ≤ ''d'' ≤ ''b''}}, one has

<math display="block">
\Pr\left( X_I \in [c,d]\right) = \frac{d - c}{b - a}
</math>

where the last equality results from the [[Probability axioms#Unitarity|unitarity axiom]] of probability.  The [[probability density function]] of a CURV <math>X \sim \operatorname {U}[a, b]</math> is given by the [[indicator function]] of its interval of [[Support (mathematics)|support]] normalized by the interval's length: <math display="block">f_X(x) = \begin{cases} 
 \displaystyle{1 \over b-a}, & a \le x \le b \\ 
  0, & \text{otherwise}. 
\end{cases}</math>Of particular interest is the uniform distribution on the [[unit interval]] <math>[0, 1]</math>.  Samples of any desired [[probability distribution]] <math>\operatorname{D}</math> can be generated by calculating the [[quantile function]] of <math>\operatorname{D}</math> on a [[Random number generation|randomly-generated number]] distributed uniformly on the unit interval.  This exploits [[Cumulative distribution function#Properties|properties of cumulative distribution functions]], which are a unifying framework for all random variables.