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 $\mathit{L}\mathit{o}\mathit{g}\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x;m,s)=\{\begin{array}{ll}{\displaystyle \frac{1}{xs\sqrt{2\pi }}\exp (-{\displaystyle \frac{{(\log x-m)}^2}{2s^2}})}& \text{for }x\ge 0\\ \\ 0& \text{for }x<0\end{array}$$

Categories: Package distrib ·

Function: cdf_lognormal (x,m,s) ¶

Returns the value at x of the distribution function of a $\mathit{L}\mathit{o}\mathit{g}\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}(m,s)$ random variable, with $s>0$. This function is defined in terms of Maxima’s built-in error function erf.

The cdf is

$$F(x;m,s)=\{\begin{array}{ll}{\displaystyle \frac{1}{2}[1+\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{\log x-m}{s\sqrt{2}}\right)]}& \text{for }x>0\\ \\ 0& \text{for }x\le 0\end{array}$$

(%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.

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Function: quantile_lognormal (q,m,s) ¶

Returns the q-quantile of a $\mathit{L}\mathit{o}\mathit{g}\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}(m,s)$ 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

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Function: mean_lognormal (m,s) ¶

Returns the mean of a $\mathit{L}\mathit{o}\mathit{g}\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X]=\exp (m+\frac{s^2}{2})$$

Categories: Package distrib ·

Function: var_lognormal (m,s) ¶

Returns the variance of a $\mathit{L}\mathit{o}\mathit{g}\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The variance is

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

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Function: std_lognormal (m,s) ¶

Returns the standard deviation of a $\mathit{L}\mathit{o}\mathit{g}\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X]=\sqrt{(\exp \left(s^2\right)-1)}\exp (m+\frac{s^2}{2})$$

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Function: skewness_lognormal (m,s) ¶

Returns the skewness coefficient of a $\mathit{L}\mathit{o}\mathit{g}\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

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

Categories: Package distrib ·

Function: kurtosis_lognormal (m,s) ¶

Returns the kurtosis coefficient of a $\mathit{L}\mathit{o}\mathit{g}\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

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

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

Returns a $\mathit{L}\mathit{o}\mathit{g}\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}(m,s)$ 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 ·