In mathematics, the term integrable function refers to a function whose integral may be calculated. Unless qualified, the integral in question is usually the Lebesgue integral. Otherwise, one can say that the function is "Riemann integrable" (i.e., its Riemann integral exists), "Henstock-Kurzweil integrable," etc. Below we will only examine the concept of Lebesgue integrability.
Given a measurable space X with sigma-algebra σ and measure μ, a real valued function f:X → R is integrable or if bothf + and f - are measurable functions with finite Lebesgue integral. Let
f+ = max(f,0)
and
f- = max( - f,0)
be the "positive" and "negative" part of f. If f is integrable, then its integral is defined as
For a real number p ≥ 0, the function f is p-integrable if the function | f | p is integrable.
In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals.
A probability density function is everywhere non-negative and its integral from −∞ to +∞ is equal to 1.
In the field of statistical physics, a non-formal reformulation of the relation above between the derivative of the cumulative distribution function and the probability density function is generally used as the definition of the probability density function.
It is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for probability density functions and probability mass functions.
Every cumulative distribution function F is (not necessarily strictly) monotone increasing and continuous from the right (right-continuous).
The Kolmogorov-Smirnov test is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution.
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