Question

The PDF of a Gaussian Random Variable is given as :

$f_x\left(x\right)=\frac{1}{\sqrt{32\pi }}{c}^{-\frac{(x-5{)}^2}{32}}$

Which of the following options is/are correct?

• A. $F_X\left(0\right)=1-Q(-1.25)$
• B. $P(X\ge 5)=1/2$
• C. $P(X\ge 0)=P(X\le 0)$
• D. $F_x\left(5\right)=1/2$

Answer 1

Answer: (A), (B), (D)

$F_x\left(5\right)=1/2$

Comparing with standard equation,

$f_x\left(x\right)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{c}^{\frac{-(x-\mu {)}^2}{2{\sigma }^2}}$

We get $\mu =5;{\sigma }^2=16$

$\left(a\right)\therefore P(X\ge 0)=1-P(X<0)\ne P(X\le 0)$

$\left(b\right)\Rightarrow P(X\ge 5)=1/2$

$\left(c\right)\Rightarrow F_x\left(x\right)=1-Q\left(\frac{x-\mu }{\sigma }\right)=1-Q\left(\frac{x-5}{4}\right)$

$\therefore F_x\left(0\right)=1-Q(-1.25)$

$\left(d\right)F_x\left(5\right)=1-Q\left(0\right)=1-0.5=0.5=1/2$