Editing Normal distribution (section)

=== Alternative parameterizations ===
Some authors advocate using the [[precision (statistics)|precision]] {{tmath|\tau}} as the parameter defining the width of the distribution, instead of the standard deviation {{tmath|\sigma}} or the variance {{tmath|\sigma^2}}. The precision is normally defined as the reciprocal of the variance, {{tmath|1/\sigma^2}}.<ref>{{harvtxt |Bernardo |Smith |2000 |page=121 }}</ref> The formula for the distribution then becomes

<math display=block>f(x) = \sqrt{\frac\tau{2\pi}} e^{-\tau(x-\mu)^2/2}.</math>

This choice is claimed to have advantages in numerical computations when {{tmath|\sigma}} is very close to zero, and simplifies formulas in some contexts, such as in the [[Bayesian statistics|Bayesian inference]] of variables with [[multivariate normal distribution]].

Alternatively, the reciprocal of the standard deviation <math display=inline>\tau'=1/\sigma</math> might be defined as the ''precision'', in which case the expression of the normal distribution becomes

<math display=block>f(x) = \frac{\tau'}{\sqrt{2\pi}} e^{-(\tau')^2(x-\mu)^2/2}.</math>

According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the [[quantile]]s of the distribution.

Normal distributions form an [[exponential family]] with [[natural parameter]]s <math display=inline>\textstyle\theta_1=\frac{\mu}{\sigma^2}</math> and <math display=inline>\textstyle\theta_2=\frac{-1}{2\sigma^2}</math>, and natural statistics ''x'' and ''x''<sup>2</sup>. The dual expectation parameters for normal distribution are {{math|1=''η''<sub>1</sub> = ''μ''}} and {{math|1=''η''<sub>2</sub> = ''μ''<sup>2</sup> + ''σ''<sup>2</sup>}}.