How to prove using the transformation method, the standardisation of a normal variable.

Question

Basically I'd like to know how to prove, using the transformation method, that given a non-standard normally distributed random variable $X$ that follows $N(μ, σ^2)$, If $Z = \frac{(X-μ)}{σ}$ then $Z$ is follows a standard normal distribution.

Answer 1

This only works continuous and one-to-one functions.

1. Write PDF of $X$
2. Find the inverse function $X = g^{−1}(Y)$ and verify that it is continuous and one-to-one for all $Y ∈ {g(x)|x ∈ X}$
3. If the derivative of $g^{-1}(Y)$ exists and is not zero: $f_Y(y) = f_X (g^{-1}(y))* |\frac{d[g^{-1}(y)]}{dy}|$