Observe P[Z≤z]=P[X−μσ≤z]=P[X≤σz+μ]. Substitute this into the cumulative distribution function of the normal distribution to obtain ∫σz+μ−∞1σ√2πexp(−12(x−μσ)2)dx. We will need to use the substitution t=x−μσ. Rearranging for x gives x=σt+μ. Differentiating with respect to t gives dxdt=σ. Substituting this into the above integral yields ∫z−∞1σ√2πexp(−12t2)σdt (note how the upper limit has changed as well here - see integration by substitution). Lastly cancelling σ gives ∫z−∞1√2πexp(−12t2)dt which is the cumulative distribution function of the standard normal distribution. We will call this function Φ(z).