Noun
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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
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t
)
=
P
r
(
X
≤
t
)
{\displaystyle F_{X}(t)=\mathrm {Pr} (X\leq t)}
.
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. Source: Internet
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. Source: Internet
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. Source: Internet
Multivariate case When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. Source: Internet
Note on terminology: some authors use the term "continuous distribution" to denote the distribution with continuous cumulative distribution function. Source: Internet
The convergence in the Central limit theorem is uniform because the limiting cumulative distribution function is continuous. Source: Internet