Cumulative distribution function: Difference between revisions
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as long as the derivative exists. | as long as the derivative exists. | ||
The CDF of | The CDF of an [[absolutely continuous random variable]] <math>X</math> can be expressed as the integral of its probability density function <math>f_X</math> as follows:<ref name="KunIlPark" />{{rp|p=86}} | ||
<math display="block">F_X(x) = \int_{-\infty}^x f_X(t) \, dt.</math> | <math display="block">F_X(x) = \int_{-\infty}^x f_X(t) \, dt.</math> | ||
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== Properties == | == Properties == | ||
[[File:Discrete probability distribution illustration.svg|right|thumb|From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part | [[File:Discrete probability distribution illustration.svg|right|thumb|From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part]] | ||
[[File:Discrete probability distribution with a countable set of discontinuities.svg|right|thumb|Example of a cumulative distribution function with a countably infinite set of discontinuities | [[File:Discrete probability distribution with a countable set of discontinuities.svg|right|thumb|Example of a cumulative distribution function with a countably infinite set of discontinuities]] | ||
Every cumulative distribution function <math>F_X</math> is [[monotone increasing|non-decreasing]]<ref name=KunIlPark/>{{rp|p | Every cumulative distribution function <math>F_X</math> is [[monotone increasing|non-decreasing]]<ref name=KunIlPark/>{{rp|p=78}} and [[right-continuous]],<ref name=KunIlPark/>{{rp|p=79}} which makes it a [[càdlàg]] function. Furthermore, | ||
<math display="block">\lim_{x \to -\infty} F_X(x) = 0, \quad \lim_{x \to +\infty} F_X(x) = 1.</math> | <math display="block">\lim_{x \to -\infty} F_X(x) = 0, \quad \lim_{x \to +\infty} F_X(x) = 1.</math> | ||