#### 52.2.8 Lognormal Random Variable

The lognormal distribution is distribution for a random variable whose logarithm is normally distributed.

Function: pdf_lognormal (x,m,s)

  Returns the value at x of the density function of a \({\it Lognormal}(m,s)\) random variable, with s>0. To make use of this function, write first `load("distrib")`.

  The pdf is $$ f(x; m, s) = \cases{ \displaystyle{1\over x s \sqrt{2\pi}} \exp\left(-\displaystyle{\left(\log x - m\right)^2\over 2s^2}\right) & for $x \ge 0$ \cr \cr 0 & for $x < 0$ } $$

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Function: cdf_lognormal (x,m,s)

  Returns the value at x of the distribution function of a \({\it Lognormal}(m,s)\) random variable, with s>0. This function is defined in terms of Maxima’s built-in error function `erf`.

  The cdf is $$ F(x; m, s) = \cases{ \displaystyle{1\over 2}\left[1+{\rm erf}\left({\log x - m\over s\sqrt{2}}\right)\right] & for $x > 0$ \cr \cr 0 & for $x \le 0$ } $$

      (%i1) load ("distrib")$

      (%i2) cdf_lognormal(x,m,s);
                                       log(x) - m
                                   erf(----------)
                                       sqrt(2) s     1
      (%o2)          unit_step(x) (--------------- + -)
                                          2          2

  See also `erf`.

  Categories:Package distrib ·

Function: quantile_lognormal (q,m,s)

  Returns the q-quantile of a \({\it Lognormal}(m,s)\) random variable, with s>0; in other words, this is the inverse of `cdf_lognormal`. Argument q must be an element of [0,1]. To make use of this function, write first `load("distrib")`.

      (%i1) load ("distrib")$

      (%i2) quantile_lognormal(95/100,0,1);
                           sqrt(2) inverse_erf(9/10)
      (%o2)              %e

      (%i3) float(%);
      (%o3)                   5.180251602233015

  Categories:Package distrib ·

Function: mean_lognormal (m,s)

  Returns the mean of a \({\it Lognormal}(m,s)\) random variable, with s>0. To make use of this function, write first `load("distrib")`.

  The mean is $$ E[X] = \exp\left(m+{s^2\over 2}\right) $$

  Categories:Package distrib ·

Function: var_lognormal (m,s)

  Returns the variance of a \({\it Lognormal}(m,s)\) random variable, with s>0. To make use of this function, write first `load("distrib")`.

  The variance is $$ V[X] = \left(\exp\left(s^2\right) - 1\right) \exp\left(2m+s^2\right) $$

  Categories:Package distrib ·

Function: std_lognormal (m,s)

  Returns the standard deviation of a \({\it Lognormal}(m,s)\) random variable, with s>0. To make use of this function, write first `load("distrib")`.

  The standard deviation is $$ D[X] = \sqrt{\left(\exp\left(s^2\right) - 1\right)} \exp\left(m+{s^2\over 2}\right) $$

  Categories:Package distrib ·

Function: skewness_lognormal (m,s)

  Returns the skewness coefficient of a \({\it Lognormal}(m,s)\) random variable, with s>0. To make use of this function, write first `load("distrib")`.

  The skewness coefficient is $$ SK[X] = \left(\exp\left(s^2\right)+2\right)\sqrt{\exp\left(s^2\right)-1} $$

  Categories:Package distrib ·

Function: kurtosis_lognormal (m,s)

  Returns the kurtosis coefficient of a \({\it Lognormal}(m,s)\) random variable, with s>0. To make use of this function, write first `load("distrib")`.

  The kurtosis coefficient is $$ KU[X] = \exp\left(4s^2\right)+2\exp\left(3s^2\right)+3\exp\left(2s^2\right)-3 $$

  Categories:Package distrib ·

Function: random_lognormal (m,s)
`random_lognormal` (m,s,n)

  Returns a \({\it Lognormal}(m,s)\) random variate, with s>0. Calling `random_lognormal` with a third argument n, a random sample of size n will be simulated.

  Log-normal variates are simulated by means of random normal variates. See `random_normal` for details.

  To make use of this function, write first `load("distrib")`.

  Categories:Package distrib ·Random numbers ·