Documentation

# logncdf

Lognormal cumulative distribution function

## Syntax

``p = logncdf(x)``
``p = logncdf(x,mu)``
``p = logncdf(x,mu,sigma)``
``[p,pLo,pUp] = logncdf(x,mu,sigma,pCov)``
``[p,pLo,pUp] = logncdf(x,mu,sigma,pCov,alpha)``
``___ = logncdf(___,'upper')``

## Description

````p = logncdf(x)` returns the cumulative distribution function (cdf) of the standard lognormal distribution, evaluated at the values in `x`. In the standard lognormal distribution, the mean and standard deviation of logarithmic values are 0 and 1, respectively.```
````p = logncdf(x,mu)` returns the cdf of the lognormal distribution with the distribution parameters `mu` (mean of logarithmic values) and 1 (standard deviation of logarithmic values), evaluated at the values in `x`.```

example

````p = logncdf(x,mu,sigma)` returns the cdf of the lognormal distribution with the distribution parameters `mu` (mean of logarithmic values) and `sigma` (standard deviation of logarithmic values), evaluated at the values in `x`.```

example

````[p,pLo,pUp] = logncdf(x,mu,sigma,pCov)` also returns the 95% confidence bounds [`pLo`,`pUp`] of `p` using the estimated parameters (`mu` and `sigma`) and their covariance matrix `pCov`.```
````[p,pLo,pUp] = logncdf(x,mu,sigma,pCov,alpha)` specifies the confidence level for the confidence interval `[pLo,pUp]` to be `100(1–alpha)`%.```

example

````___ = logncdf(___,'upper')` returns the complement of the cdf, evaluated at the values in `x`, using an algorithm that more accurately computes the extreme upper-tail probabilities. `'upper'` can follow any of the input argument combinations in the previous syntaxes.```

## Examples

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Compute the cdf values evaluated at the values in `x` for the lognormal distribution with mean `mu` and standard deviation `sigma`.

```x = 0:0.2:10; mu = 0; sigma = 1; p = logncdf(x,mu,sigma);```

Plot the cdf.

```plot(x,p) grid on xlabel('x') ylabel('p')``` Find the maximum likelihood estimates (MLEs) of the lognormal distribution parameters, and then find the confidence interval of the corresponding cdf value.

Generate 1000 random numbers from the lognormal distribution with the parameters 5 and 2.

```rng('default') % For reproducibility n = 1000; % Number of samples x = lognrnd(5,2,n,1);```

Find the MLEs for the distribution parameters (mean and standard deviation of logarithmic values) by using `mle`.

`phat = mle(x,'distribution','LogNormal')`
```phat = 1×2 4.9347 1.9969 ```
```muHat = phat(1); sigmaHat = phat(2);```

Estimate the covariance of the distribution parameters by using `lognlike`. The function `lognlike` returns an approximation to the asymptotic covariance matrix if you pass the MLEs and the samples used to estimate the MLEs.

`[~,pCov] = lognlike(phat,x)`
```pCov = 2×2 0.0040 -0.0000 -0.0000 0.0020 ```

Find the cdf value at 0.5 and its 95% confidence interval.

`[p,pLo,pUp] = logncdf(0.5,muHat,sigmaHat,pCov)`
```p = 0.0024 ```
```pLo = 0.0016 ```
```pUp = 0.0037 ```

`p` is the cdf value of the lognormal distribution with the parameters `muHat` and `sigmaHat`. The interval `[pLo,pUp]` is the 95% confidence interval of the cdf evaluated at 0.5, considering the uncertainty of `muHat` and `sigmaHat` using `pCov`. The 95% confidence interval means the probability that `[pLo,pUp]` contains the true cdf value is 0.95.

Determine the probability that an observation from a standard lognormal distribution will fall on the interval `[exp(10),Inf]`.

`p1 = 1 - logncdf(exp(10))`
```p1 = 0 ```

`logncdf(exp(10))` is nearly 1, so `p1` becomes 0. Specify `'upper'` so that `logncdf` computes the extreme upper-tail probabilities more accurately.

`p2 = logncdf(exp(10),'upper')`
```p2 = 7.6199e-24 ```

You can also use `'upper'` to compute a right-tailed p-value.

## Input Arguments

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Values at which to evaluate the cdf, specified as a positive scalar value or an array of positive scalar values.

If you specify `pCov` to compute the confidence interval `[pLo,pUp]`, then `x` must be a scalar value.

To evaluate the cdf at multiple values, specify `x` using an array. To evaluate the cdfs of multiple distributions, specify `mu` and `sigma` using arrays. If one or more of the input arguments `x`, `mu`, and `sigma` are arrays, then the array sizes must be the same. In this case, `logncdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `p` is the cdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `x`.

Example: `[-1,0,3,4]`

Data Types: `single` | `double`

Mean of logarithmic values for the lognormal distribution, specified as a scalar value or an array of scalar values.

If you specify `pCov` to compute the confidence interval `[pLo,pUp]`, then `mu` must be a scalar value.

To evaluate the cdf at multiple values, specify `x` using an array. To evaluate the cdfs of multiple distributions, specify `mu` and `sigma` using arrays. If one or more of the input arguments `x`, `mu`, and `sigma` are arrays, then the array sizes must be the same. In this case, `logncdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `p` is the cdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `x`.

Example: `[0 1 2; 0 1 2]`

Data Types: `single` | `double`

Standard deviation of logarithmic values for the lognormal distribution, specified as a positive scalar value or an array of positive scalar values.

If you specify `pCov` to compute the confidence interval `[pLo,pUp]`, then `sigma` must be a scalar value.

To evaluate the cdf at multiple values, specify `x` using an array. To evaluate the cdfs of multiple distributions, specify `mu` and `sigma` using arrays. If one or more of the input arguments `x`, `mu`, and `sigma` are arrays, then the array sizes must be the same. In this case, `logncdf` expands each scalar input into a constant array of the same size as the array inputs. Each element in `p` is the cdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `x`.

Example: `[1 1 1; 2 2 2]`

Data Types: `single` | `double`

Covariance of the estimates `mu` and `sigma`, specified as a 2-by-2 matrix.

If you specify `pCov` to compute the confidence interval `[pLo,pUp]`, then `x`, `mu`, and `sigma` must be scalar values.

You can estimate the maximum likelihood estimates of `mu` and `sigma` by using `mle`, and estimate the covariance of `mu` and `sigma` by using `lognlike`. For an example, see Confidence Interval of Lognormal cdf Value.

Data Types: `single` | `double`

Significance level for the confidence interval, specified as a scalar in the range (0,1). The confidence level is `100(1–alpha)`%, where `alpha` is the probability that the confidence interval does not contain the true value.

Example: `0.01`

Data Types: `single` | `double`

## Output Arguments

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cdf values, evaluated at the values in `x`, returned as a scalar value or an array of scalar values. `p` is the same size as `x`, `mu`, and `sigma` after any necessary scalar expansion. Each element in `p` is the cdf value of the distribution specified by the corresponding elements in `mu` and `sigma`, evaluated at the corresponding element in `x`.

Lower confidence bound for `p`, returned as a scalar value or an array of scalar values. `pLo` has the same size as `p`.

Upper confidence bound for `p`, returned as a scalar value or an array of scalar values. `pUp` has the same size as `p`.

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### Lognormal Distribution

The lognormal distribution is a probability distribution whose logarithm has a normal distribution.

The cumulative distribution function (cdf) of the lognormal distribution is

`$p=F\left(x|\mu ,\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}{\int }_{0}^{x}\frac{1}{t}\mathrm{exp}\left\{\frac{-{\left(\mathrm{log}t-\mu \right)}^{2}}{2{\sigma }^{2}}\right\}dt,\text{ }\text{for}\text{\hspace{0.17em}}x>0.$`

## Algorithms

• The `logncdf` function uses the complementary error function `erfc`. The relationship between `logncdf` and `erfc` is

`$\text{logncdf}\left(x,0,1\right)=\frac{1}{2}\text{erfc}\left(-\frac{\mathrm{log}x}{\sqrt{2}}\right).$`

The complementary error function `erfc(x)` is defined as

`$\text{erfc}\left(x\right)=1-\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_{x}^{\infty }{e}^{-{t}^{2}}dt.$`

• The `logncdf` function computes confidence bounds for `p` by using the delta method. The normal distribution cdf value of `log(x)` with the parameters `mu` and `sigma` is equivalent to the cdf value of `(log(x)–mu)/sigma` with the parameters 0 and 1. Therefore, the `logncdf` function estimates the variance of `(log(x)–mu)/sigma` using the covariance matrix of `mu` and `sigma` by the delta method, and finds the confidence bounds of `(log(x)–mu)/sigma` using the estimates of this variance. Then, the function transforms the bounds to the scale of `p`. The computed bounds give approximately the desired confidence level when you estimate `mu`, `sigma`, and `pCov` from large samples.

## Alternative Functionality

• `logncdf` is a function specific to lognormal distribution. Statistics and Machine Learning Toolbox™ also offers the generic function `cdf`, which supports various probability distributions. To use `cdf`, create a `LognormalDistribution` 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 `logncdf` is faster than the generic function `cdf`.

• Use the Probability Distribution Function app to create an interactive plot of the cumulative distribution function (cdf) or probability density function (pdf) for a probability distribution.

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

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