How do I derive that $(\frac{X-\mu}{\sigma})$ is $N(0,1)$?

Question

Suppose that $X$ is $N(\mu,\sigma^2)$ distributed. I have only learned the definition of normal distributions and I was told the fact that $(\frac{X-\mu}{\sigma})$ is $N(0,1)$.

But how can I get a proof of it? Would anyone help?

Answer 1

Hint:

By finding mean and variance of $\frac{X-\mu}{\sigma}$ (which are completely determining for the distribution of normally distributed random variables).

For this use linearity of expectation and use equalities like: $$\mathsf{Var}(aX+b)=a^2\mathsf{Var}X$$