If $Y\sim N(\mu T,\sigma^2 T)$ then $Y=\mu T+\sigma \sqrt{T} Z$ where $Z\sim N(0,1)$

Question

Let $S_T$ be the price of a traded asset at time $T$. Also let:

$\ln(\frac{S_T}{S_0}) \sim N(\mu T,\sigma^2 T)$

My question is, how is it that: $\ln(\frac{S_T}{S_0})=\mu T+\sigma \sqrt{T} Z$ where $Z\sim N(0,1)$ ?

From the distribution fact, how is $\ln(\frac{S_T}{S_0})$ expressed the way it is ?

Answer 1

Hint: We know, that if $X \sim N(\mu,\sigma^2),$ then $\forall a,b \in \mathbb R$ the variable $aX + b$ is also normally distributed. So now count the mean and variance of random variable $$ \frac{1}{\sigma\sqrt T} \ln \left( \frac{S_T}{S_0} \right) - \mu T.$$