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Converting from the Normal Distribution to the Standard Normal Distribution

Observe P[Zz]=P[Xμσz]=P[Xσz+μ]. Substitute this into the cumulative distribution function of the normal distribution to obtain σz+μ1σ2πexp(12(xμσ)2)dx. We will need to use the substitution t=xμσ. Rearranging for x gives x=σt+μ. Differentiating with respect to t gives dxdt=σ. Substituting this into the above integral yields z1σ2πexp(12t2)σdt (note how the upper limit has changed as well here - see integration by substitution). Lastly cancelling σ gives z12πexp(12t2)dt which is the cumulative distribution function of the standard normal distribution. We will call this function Φ(z).

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