Difference between revisions of "Cumulative distribution function"
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Let $\mathbb{T}$ be a [[time scale]]. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be a [[probability density function]]. The following function is called the cumulative distribution function (or cdf) of $f$: | Let $\mathbb{T}$ be a [[time scale]]. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be a [[probability density function]]. The following function is called the cumulative distribution function (or cdf) of $f$: | ||
$$F(x) = \displaystyle\int_0^x f(t) \Delta t.$$ | $$F(x) = \displaystyle\int_0^x f(t) \Delta t.$$ | ||
| + | |||
| + | Let $f(x,y)$ be a [[joint time scales probability density function]], then we have a joint time scales cumulative distribution function by | ||
| + | $$F_{X,Y}(x,y)=P(X<x,Y<y)=\displaystyle\int_0^x\int_0^y f_{X,Y}(s,t) \Delta t \Delta s.$$ | ||
=References= | =References= | ||
[https://mospace.umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1 Probability theory on time scales and applications to finance and inequalities by Thomas Matthews] | [https://mospace.umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1 Probability theory on time scales and applications to finance and inequalities by Thomas Matthews] | ||
Latest revision as of 04:37, 6 March 2015
Let $\mathbb{T}$ be a time scale. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be a probability density function. The following function is called the cumulative distribution function (or cdf) of $f$: $$F(x) = \displaystyle\int_0^x f(t) \Delta t.$$
Let $f(x,y)$ be a joint time scales probability density function, then we have a joint time scales cumulative distribution function by $$F_{X,Y}(x,y)=P(X<x,Y<y)=\displaystyle\int_0^x\int_0^y f_{X,Y}(s,t) \Delta t \Delta s.$$
References
Probability theory on time scales and applications to finance and inequalities by Thomas Matthews
