Change of mean for a standard normal variable Beginning to learn a bit of probability theory and I'm a little confused with this one. I know that given a Standard Normal variable X
$X \sim \mathcal{N}(0, 1)$, if $Y = X + c$ then $Y \sim \mathcal{N}(c, \sigma^2)$
Now if, 
$Y = aX+b $, is $Y \sim \mathcal{N}($b/a$, \sigma^2)$?

Let $X \sim \mathcal{N}(0, 1)$. Then, $E[X+c] = E[X]  + E[c] = E[X] + c = c$. Also, $\text{Var}[X+c] = \text{Var}[X] = 1. $ So $Y \sim \mathcal{N}(c,1)$.
If instead we have $Y=aX+b$, then $E[Y] = E[aX+b] = E[aX] + E[b] = a E[X] + b = a \cdot 0 + b$. Also, $\text{Var}[aX+b] = \text{Var}[aX] = a^2 \text{Var}[X] = a^2 $. So $Y \sim\mathcal{N}(b, a^2 )$.
Take a look at the properties of expectation/variance, they will come up alot.
https://en.wikipedia.org/wiki/Expected_value#Properties
https://en.wikipedia.org/wiki/Variance#Basic_properties
EDIT: I'm assuming that you know these transformations still result in a normal distribution and simply checking their mean & variance. The other answer provided is much more thorough . . . 
