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Say we now a Random Variable X is normal $X \sim N(\mu, \sigma) $

Then we know that :

$Z= (X-\mu)/\sigma $ ~$N(0,1)$

I answered in another question that we know Y is normal if it can be obtained through the inverse transform $Z^{-1}$ , inverse to above, i.e., X is normal iff:

$X== \sigma Z + \mu $ ,

for some Real $\sigma >0, \mu $

Is this correct?

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If I've understood your question...

If $Z$ is a standard normal random variable $Z \sim \mathcal{N}(0,1)$

Then $X = \mu + \sigma Z $ is a normal random variable $X \sim \mathcal{N}(\mu, \sigma^2)$.

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Based on OP's "iff", I think we need to prove that:

Given $Z$ follows standard normal distribution,

  1. For real numbers $(\mu, \sigma) \in R^2$, $Y=\mu + \sigma Z$ follows normal distribution, as given by Demetri Pananos.

  2. If $Y$ follows normal distribution, then there exist real numbers $(\mu, \sigma) \in R^2$ such that $Y=\mu + \sigma Z$. It is true because $\left(E(Y), \sqrt{Var(Y)}\right)$ meet the requirement.

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