User:Egm6322.s12.team2.Xia/RP13.4

= R 13.4 - Error function =

Given
Definiation of error function

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined as:

$$\displaystyle \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt. $$

(When x is negative, the integral is interpreted as the negative of the integral from x to zero.) (13.4.1) From the (1)p.48-33 and (2)p.48-33

$$\displaystyle F_{X}(x)=\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right] $$ (13.4.2)

$$\displaystyle f_{X}(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(x-\mu)^{2}}{\sigma\sqrt{2}}\right] $$ (13.4.3)

Find
Show that 13.4.2 can be obtained from 13.4.3 and vice versa.

Solution
$$\displaystyle $$ (13.4.4)

$$\displaystyle $$ (13.4.4)

$$\displaystyle $$ (13.4.4)

$$\displaystyle $$ (13.4.4)

F_{X}(x)=\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]

can be obtained from (2)p.48-33

f_{X}(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(x-\mu)^{2}}{\sigma\sqrt{2}}\right]

and vice versa.