Question

A gaussian random variable X has 𝜇𝑥 = 2, and 𝜎𝑥 = 2 . (a) Find P(X > 1.0). (b) Find P(X < −1.0).

Answer 1

The probability density function is: $f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}$ with $\mu=2$ and $\sigma=2$.

a). $P(X>1.0)=\int_1^{+\infty}\frac{1}{2\sqrt{2\pi}}e^{-\frac12\left(\frac{x-2}{2}\right)^2}dx\approx0.6915$

b). $P(X<-1.0)=\int_{-\infty}^{-1}\frac{1}{2\sqrt{2\pi}}e^{-\frac12\left(\frac{x-2}{2}\right)^2}dx\approx0.067$

The following sequence of commands in Maple was used:

restart;

s:=2;u:=2;

f:=1/s/sqrt(2*Pi)*exp(-1/2*((x-u)/s)^2);

evalf(int(f,x=1..+infinity));

evalf(int(f,x=-infinity..-1));