Normal distribution function

Question

Given Data :

Normal Distribution function

$$P(x \leq -1) = P(y \geq 2)$$

x : mean= 1 variance= 4

y : mean= -1 variance= to be found..?.

Ans: 9

How to solve..?

Answer 1

Firstly, $P(x\le-1)=\Phi\left(\frac{-1-1}{\sqrt4}\right)=\Phi(-1)$ Also, $P(y\ge2)=P(y\le-4)=\Phi\left(\frac{-4-(-1)}{\sigma}\right)=\Phi\left(\frac{-3}{\sigma}\right)$.

So we have $\frac{-3}{\sigma}=-1$ and $\sigma^2=9$.