In probability theory, the<unk>  <unk> , also called probability<unk><unk>  or just<unk><unk> , completely describes the probability<unk>  of a real-valued random variable X. For every real number x, the<unk>  of X is given by
where the right-hand side represents the probability that the random variable X takes on a value less than or
equal to x. The probability that X lies in the interval (a, b] is therefore F(b) − F(a) if a < b.  It is conventional to use a capital F for a<unk> , in contrast to the lower-case f used for probability density<unk> s and probability mass<unk> s.
The<unk>  of X can be defined in terms of the probability density<unk>  f as follows:
Note that in the definition above, the "less or equal" sign, '≤' is a convention, but it is
an important and universally used one. The proper use of tables of the Binomial and Poisson<unk> s depend upon this convention. Moreover, important formulas like Levy's inversion formula for the characteristic<unk>  also rely on the "less or equal" formulation.
== Properties ==leftright|thumb|From top to bottom, the<unk>  of a discrete probability<unk> , continuous probability<unk> , and a<unk>  which has both a continuous part and a discrete part.]]
Every<unk>  F is (not necessarily strictly) monotone increasing and right-continuous. Furthermore, we have
Every<unk>  with these four properties is a<unk> . The properties imply that all<unk> s are càdlàg<unk> s.
If X is a discrete random variable, then it attains values x1, x2, ... with probability pi = P(xi), and the<unk>  of X will be discontinuous at the points xi and constant in between:
If the<unk>  F of X is continuous, then X is a continuous random variable; if furthermore F is absolutely continuous, then there exists a Lebesgue-integrable<unk>  f(x) such that
for all real numbers a and b.  (The first of the two equalities displayed above would not be correct in general if we had not said that the<unk>  is continuous.  Continuity of the<unk>  implies that P(X = a) = P(X = b) = 0, so the difference between "<" and "≤" ceases to be important in this context.)  The<unk>  f is equal to the derivative of F almost everywhere, and it is called the probability density<unk>  of the<unk>  of X.
The "point probability" that X is exactly b can be found as
Cumulative distribution function
Ancillary statistic
Covariance matrix
Cohen's kappa
Degrees of freedom 
Quantum yield
Sample size
Bar chart