cumulative distribution function
Noun
Meaning
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cumulative distribution function (plural cumulative distribution functions)
A function which at each point t of the sample space has as its value the probability that a given random variable is less than (or equal) t. In symbols,
F
X
(
t
)
=
P
r
(
X
≤
t
)
{\displaystyle F_{X}(t)=\mathrm {Pr} (X\leq t)}
.
Examples
Alternatively, for a cumulative distribution function F(x) with inverse x(F), the Lorenz curve L(F) is directly given by: : The inverse x(F) may not exist because the cumulative distribution function has intervals of constant values.
Examples * As an example, suppose we have a random variable and a cumulative distribution function : In order to perform an inversion we want to solve for : : From here we would perform steps one, two and three.
Furthermore, : Every function with these four properties is a CDF, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable.
Multivariate case When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined.
Note on terminology: some authors use the term "continuous distribution" to denote the distribution with continuous cumulative distribution function.
The convergence in the Central limit theorem is uniform because the limiting cumulative distribution function is continuous.