Normal (univariate)

Parameters

Mean \(mu\), Variance \(\sigma ^{2}\)

Kernel

\(\sigma ^{-1}\exp \left( -\dfrac{\left( x-\mu \right) ^{2}}{2\sigma ^{2}}\right) \)

Support

\(\left( -\infty ,\infty \right) \)

Mean

\(\mu\)

Variance

\(\sigma ^{2}\)

Main Uses

Distribution of univariate error processes. Asymptotic distributions. Prior, exact and approximate posteriors for parameters with unlimited ranges.

Density Function

%DENSITY(x) is the (non-logged) standard Normal density.

%LOGDENSITY(variance,u) is, more generally, the log density. Use %LOGDENSITY(sigmasq,x-mu) to compute

\(\log f\left( x|\mu ,\sigma^{2}\right) \)

CDF

%CDF(x) is the standard Normal CDF. Use %CDF((x-mu)/sigma) to get \(\mathbf{F}(x|\mu ,\sigma ^{2})\)

%LOGCDF(v,x) is the log of the CDF of \({x/\sqrt v }\) where v is the variance (mean zero assumed).

%ZTEST(z) gives the two-tailed tail probability (probability a N(0,1) exceeds z in absolute value).

Inverse CDF

%INVNORMAL(p) is the standard Normal inverse CDF

%INVZTEST(p) is the two-tailed critical value for tail probability p.

Random Draws

%RAN(s) draws one or more (depending upon the target) independent \(N\left( 0,s^{2}\right)\).

%RANMAT(m,n) draws a matrix of independent N(0,1).