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# logninv

Lognormal inverse cumulative distribution function

## Syntax

X = logninv(P,mu,sigma)
[X,XLO,XUP] = logninv(P,mu,sigma,pcov,alpha)

## Description

X = logninv(P,mu,sigma) returns values at P of the inverse lognormal cdf with distribution parameters mu and sigma. mu and sigma are the mean and standard deviation, respectively, of the associated normal distribution. mu and sigma can be vectors, matrices, or multidimensional arrays that all have the same size, which is also the size of X. A scalar input for P, mu, or sigma is expanded to a constant array with the same dimensions as the other inputs.

[X,XLO,XUP] = logninv(P,mu,sigma,pcov,alpha) returns confidence bounds for X when the input parameters mu and sigma are estimates. pcov is the covariance matrix of the estimated parameters. alpha specifies 100(1 - alpha)% confidence bounds. The default value of alpha is 0.05. XLO and XUP are arrays of the same size as X containing the lower and upper confidence bounds.

logninv computes confidence bounds for P using a normal approximation to the distribution of the estimate

$\stackrel{^}{\mu }+\stackrel{^}{\sigma }q$

where q is the Pth quantile from a normal distribution with mean 0 and standard deviation 1. The computed bounds give approximately the desired confidence level when you estimate mu, sigma, and pcov from large samples, but in smaller samples other methods of computing the confidence bounds might be more accurate.

The lognormal inverse function is defined in terms of the lognormal cdf as

$x={F}^{-1}\left(p|\mu ,\sigma \right)=\left\{x:F\left(x|\mu ,\sigma \right)=p\right\}$

where

$p=F\left(x|\mu ,\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}{\int }_{0}^{x}\frac{{e}^{\frac{-{\left(\mathrm{ln}\left(t\right)-\mu \right)}^{2}}{2{\sigma }^{2}}}}{t}dt$

## Examples

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### Compute the Lognormal Distribution Inverse cdf

Compute the inverse cdf of a lognormal distribution with mu = 0 and sigma = 0.5.

```p = (0.005:0.01:0.995);
crit = logninv(p,1,0.5);
```

Plot the inverse cdf.

```figure;
plot(p,crit)
xlabel('Probability');
ylabel('Critical Value');
grid
```