## Posts filled under #fius

The Only man you should Chase is the ice cream Van Man Wise Words from @anniedrea #fius

## An extract on #fius

The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter {\displaystyle \Phi } (phi), is the integral ( x ) = 1 2 x e t 2 / 2 d t {\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,{\rm {d}}t} In statistics one often uses the related error function, or e r f ( x ) {\displaystyle \mathrm {erf} (x)} , defined as the probability of a random variable with normal distribution of mean 0 and variance 1/2 falling in the range [ x , x ] {\displaystyle [-x,x]} ; that is erf ( x ) = 2 0 x e t 2 d t {\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,{\rm {d}}t} These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions. However, many numerical approximations are known; see below. The two functions are closely related, namely ( x ) = 1 2 [ 1 + erf ( x 2 ) ] {\displaystyle \Phi (x)={\tfrac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]} For a generic normal distribution with density f {\displaystyle f} , mean {\displaystyle \mu } and deviation {\displaystyle \sigma } , the cumulative distribution function is F ( x ) = ( x ) = 1 2 [ 1 + erf ( x 2 ) ] {\displaystyle F(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\tfrac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]} The complement of the standard normal CDF, Q ( x ) = 1 ( x ) {\displaystyle Q(x)=1-\Phi (x)} , is often called the Q-function, especially in engineering texts. It gives the probability that the value of a standard normal random variable X {\displaystyle X} will exceed x {\displaystyle x} : P ( X > x ) {\displaystyle P(X>x)} . Other definitions of the Q {\displaystyle Q} -function, all of which are simple transformations of {\displaystyle \Phi } , are also used occasionally. The graph of the standard normal CDF {\displaystyle \Phi } has 2-fold rotational symmetry around the point (0,1/2); that is, ( x ) = 1 ( x ) {\displaystyle \Phi (-x)=1-\Phi (x)} . Its antiderivative (indefinite integral) is ( x ) d x = x ( x ) + ( x ) {\displaystyle \int \Phi (x)\,{\rm {d}}x=x\Phi (x)+\varphi (x)} . The cumulative distribution function (CDF) of the standard normal distribution can be expanded by Integration by parts into a series: ( x ) = 0.5 + 1 2 e x 2 / 2 [ x + x 3 3 + x 5 3 5 + + x 2 n + 1 ( 2 n + 1 ) ! ! + ] {\displaystyle \Phi (x)=0.5+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\ldots +{\frac {x^{2n+1}}{(2n+1)!!}}+\ldots \right]} where ! ! {\displaystyle !!} denotes the double factorial. As an example, the following Pascal function approximates the CDF: