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Normal inverse cumulative distribution function

`x = norminv(p)`

`x = norminv(p,mu)`

`x = norminv(p,mu,sigma)`

`[x,xLo,xUp] = norminv(p,mu,sigma,pCov)`

`[x,xLo,xUp] = norminv(p,mu,sigma,pCov,alpha)`

The

`norminv`

function uses the inverse complementary error function`erfcinv`

. The relationship between`norminv`

and`erfcinv`

is$$\text{norminv}(p)=-\sqrt{2}\text{\hspace{0.05em}}\text{erfcinv}(2p)$$

The inverse complementary error function

`erfcinv(x)`

is defined as`erfcinv(erfc(x))=x`

, and the complementary error function`erfc(x)`

is defined as$$\text{erfc}(x)=1-\text{erf}(x)=\frac{2}{\sqrt{\pi}}{\displaystyle {\int}_{x}^{\infty}{e}^{-{t}^{2}}dt}.$$

The

`norminv`

function computes confidence bounds for`x`

by using the delta method.`norminv(p,mu,sigma)`

is equivalent to`mu + sigma*norminv(p,0,1)`

. Therefore, the`norminv`

function estimates the variance of`mu + sigma*norminv(p,0,1)`

using the covariance matrix of`mu`

and`sigma`

by the delta method, and finds the confidence bounds using the estimates of this variance. The computed bounds give approximately the desired confidence level when you estimate`mu`

,`sigma`

, and`pCov`

from large samples.

`norminv`

is a function specific to normal distribution. Statistics and Machine Learning Toolbox™ also offers the generic function`icdf`

, which supports various probability distributions. To use`icdf`

, create a`NormalDistribution`

probability distribution object and pass the object as an input argument or specify the probability distribution name and its parameters. Note that the distribution-specific function`norminv`

is faster than the generic function`icdf`

.

[1] Abramowitz, M., and I. A. Stegun. *Handbook of Mathematical
Functions*. New York: Dover, 1964.

[2] Evans, M., N. Hastings, and B. Peacock. *Statistical
Distributions*. 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 1993.