Editing Normal distribution (section)

=== General normal distribution ===
Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor {{tmath|\sigma}} (the standard deviation) and then translated by {{tmath|\mu}} (the mean value):

<math display=block>
f(x \mid \mu, \sigma^2) =\frac 1 \sigma \varphi\left(\frac{x-\mu} \sigma \right)\,.
</math>

The probability density must be scaled by <math display=inline>1/\sigma</math> so that the [[integral]] is still&nbsp;1.

If {{tmath|Z}} is a [[standard normal deviate]], then <math display=inline>X=\sigma Z + \mu</math> will have a normal distribution with expected value {{tmath|\mu}} and standard deviation {{tmath|\sigma}}. This is equivalent to saying that the standard normal distribution {{tmath|Z}} can be scaled/stretched by a factor of {{tmath|\sigma}} and shifted by {{tmath|\mu}} to yield a different normal distribution, called {{tmath|X}}. Conversely, if {{tmath|X}} is a normal deviate with parameters {{tmath|\mu}} and <math display=inline>\sigma^2</math>, then this {{tmath|X}} distribution can be re-scaled and shifted via the formula <math display=inline>Z=(X-\mu)/\sigma</math> to convert it to the standard normal distribution. This variate is also called the standardized form of {{tmath|X}}.