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Lognormal cumulative distribution function

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

[p,plo,pup] = logncdf(x,mu,sigma,pcov,alpha)

[p,plo,pup] = logncdf(___,'upper')

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

returns
values at `x`

of the lognormal cdf with distribution
parameters `mu`

and `sigma`

. `mu`

and `sigma`

are
the mean and standard deviation, respectively, of the associated normal
distribution. `x`

, `mu`

, and `sigma`

can
be vectors, matrices, or multidimensional arrays that all have the
same size. A scalar input for `x`

, `mu`

,
or `sigma`

is expanded to a constant array with the
same dimensions as the other inputs.

`[p,plo,pup] = logncdf(x,mu,sigma,pcov,alpha)`

returns
confidence bounds for `p`

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. `plo`

and `pup`

are arrays
of the same size as `p`

containing the lower and
upper confidence bounds.

`[p,plo,pup] = logncdf(___,'upper')`

returns
the complement of the lognormal cdf at each value in `x`

,
using an algorithm that more accurately computes the extreme upper
tail probabilities. You can use `'upper'`

with any
of the previous syntaxes.

`logncdf`

computes confidence bounds for `p`

using
a normal approximation to the distribution of the estimate

$$\frac{x-\widehat{\mu}}{\widehat{\sigma}}$$

and then transforming those bounds to the scale of the output `p`

.
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 cdf is

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

[1] Evans, M., N. Hastings, and B. Peacock. *Statistical
Distributions*. 2nd ed., Hoboken, NJ: John Wiley &
Sons, Inc., 1993, pp. 102–105.

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