Find normal distribution parameters from cdf

Question

Given a random variable $X \sim Norm(\mu, \sigma)$, which we know that 20% of observation are less than 1.2 while 90% of observations are less than 1.4, how can we estimate the parameters $\mu, \sigma$? Hence, given $\Phi_{X}(1.2)=0.2$ and $\Phi_{X}(1.4)=0.9$, find $\mu, \sigma$. Thanks!

Answer 1

Given that, $\Phi^{-1}(0.2)=-0.842=P(z<-0.842)=0.2$, and $\Phi^{-1}(0.9)=1.282=P(z<1.282)=0.9$:

Solving the following system for $\mu, \sigma$, $(\mu+1.282\sigma=1.4, \mu-0.842\sigma=1.2)$ will return the parameters which are $\mu=1.2792$ and $\sigma=0.09416$.