Gaussian distribution with absolute value

Question

I am doing my homework about continuous random variable and Im struggling with this problem :

Given a Gaussian random variable $T(85,10)$, find $c$ satisfying $\mathbb{P}[|T| < c] = 0.9$.

Could you help me with this question? Thanks a lot in advance for your help!

Answer 1

HINT

1. Can you find some transformation $X = (T-a)/b$ so that $X \sim \mathcal{N}(0,1)$ and $b>0$?
2. Then $T = bX+a$ and your expression becomes $$ \begin{split} 0.9 &= \mathbb{P}[|T| < c] \\ &= \mathbb{P}[-c < T < c] \\ &= \mathbb{P}[-c < bX+a < c] \\ &= \mathbb{P}\left[\frac{-c-a}{b} < X < \frac{c-a}{b}\right] \\ &= \Phi\left(\frac{c-a}{b}\right) - \Phi\left(\frac{-c-a}{b}\right), \end{split} $$ where $\Phi$ is the standard normal CDF...