Linear transformations in normal distributions

Question

I am still a bit new to this topic, and was wondering if someone could check my work, it is a short exercise.

1. Find the distribution of $X = \mu + N(0,1)$

If we let $Z \sim N(0,1)$ then $X = \mu + Z$ and so $X \sim N(0+\mu,1)$

2. Find the distribution of $X = -N(0,1)$

Let $Y \sim N(0,1)$

Then $X = (-1)Y \implies X \sim N(0, 1 \times(-1^2)) \implies X \sim N(0,1)$

Was there anything wrong with my notation, arguments or anything like that? Are my answers correct?

any feedback is appreciated.

Answer 1

In general, if $X$ is a random variable, then for any $a,b\in\mathbb R$, $$\mathbb E[aX+b] = a\mathbb E[X] + b $$ and $$\operatorname{Var}(aX+b) = a^2\operatorname{Var}(X). $$ Since the distribution of a normal random variable is determined by its mean and variance, your (correct) conclusions follow from the above.