I have a random variable X, that is Normally distributed, and then I have another Random Variable Y, which is some sort of a function of X..... (e.g. Y = f(X)).
Is it possible for the Dependent Random Variable Y to follow a Normal Distribution?
If it is, then Under what conditions Y can have a normal distribution and under what conditions Y cannot have a normal distribution?
HINT Let $F_X(x) = \mathbb{P}[X \le x]$ be the CDF of $X$. Then assuming $f$ is invertible, $$ F_Y(x) = \mathbb{P}[Y \le y] = \mathbb{P}[f(X) \le y], $$ and you would like to claim somehow that this equals $$ \mathbb{P}[f^{-1}(f(X)) \le f^{-1}(y)] = \mathbb{P}[X \le f^{-1}(y)] = F_X\left(f^{-1}(y)\right). $$
But under what conditions on $f$ can you make that claim?