# Solved – Going from a normal distribution to a standard normal distribution with a change of variable

If \$X\$ follows a normal distribution with parameters \$mu\$ and \$sigma^2\$ show that \$Z = (X- mu)/sigma\$ follows a standard normal distribution. This doesn't seem to intuitive to me. We shift \$X\$ so that the top of the bell curve is over \$0\$, that bit makes sense. But then we scale the curve by a scalar, wouldn't this change the area under the curve, making it no longer a pdf since if we integrate \$Z\$ over the real line we need to get \$1\$? I guess my intiution is failing because I'm trying to imagine area along an infinite line.

Now I would like to go about proving this using the definition of the normal distribution. The definition I'm working with is \$\$f(x,mu,sigma) = frac{1}{sigma sqrt{2 pi}}e^{frac{-(x- mu)^2}{2 sigma^2}}\$\$

I thought the proof would be showing that \$\$f(frac{x-mu}{sigma},mu,sigma) = frac{1}{sqrt{2 pi}}e^{frac{-x^2}{2}}\$\$ But this didn't work, I got an ugly expression in the exponent and I couldn't see why the \$sigma\$ in the denominator would disappear at the front of the expression.

How Should I go about proving this is a fundamental way? I think their is probably an easier proof using properties such as how adding and multiplying normal distribution's by scalars modifies it. But what I'm really after is a proof using the normal distribution definition