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Normal random variables (also called Gaussian) is denoted by \({\it Normal}(m, s)\) where m is the mean and s > 0 is the standard deviation.
Returns the value at x of the density function of a
\({\it Normal}(m, s)\) random variable, with s>0. To make use of this function, write first load("distrib").
The pdf is $$ f(x; m, s) = {1\over s\sqrt{2\pi}} e^{\displaystyle -{(x-m)^2\over 2s^2}} $$
Returns the value at x of the distribution function of a
\({\it Normal}(m, s)\) random variable, with s>0. This function is defined in terms of Maxima’s built-in error function erf.
The cdf can be written analytically: $$ F(x; m, s) = {1\over 2} + {1\over 2} {\rm erf}\left(x-m\over s\sqrt{2}\right) $$
(%i1) load ("distrib")$
(%i2) cdf_normal(x,m,s);
x - m
erf(---------)
sqrt(2) s 1
(%o2) -------------- + -
2 2
See also erf.
Returns the q-quantile of a
\({\it Normal}(m, s)\) random variable, with s>0; in other words, this is the inverse of cdf_normal. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").
(%i1) load ("distrib")$
(%i2) quantile_normal(95/100,0,1);
9
(%o2) sqrt(2) inverse_erf(--)
10
(%i3) float(%); (%o3) 1.644853626951472
Returns the mean of a
\({\it Normal}(m, s)\) random variable, with s>0. To make use of this function, write first load("distrib").
The mean is $$ E[X] = m $$
Returns the variance of a
\({\it Normal}(m, s)\) random variable, with s>0. To make use of this function, write first load("distrib").
The variance is $$ V[X] = s^2 $$
Returns the standard deviation of a
\({\it Normal}(m, s)\) random variable, with s>0, namely s. To make use of this function, write first load("distrib").
The standard deviation is $$ D[X] = s $$
Returns the skewness coefficient of a
\({\it Normal}(m, s)\) random variable, with s>0. To make use of this function, write first load("distrib").
The skewness coefficient is $$ SK[X] = 0 $$
Returns the kurtosis coefficient of a
\({\it Normal}(m, s)\) random variable, with s>0, which is always equal to 0. To make use of this function, write first load("distrib").
The kurtosis coefficient is $$ KU[X] = 0 $$
Returns a
\({\it Normal}(m, s)\) random variate, with s>0. Calling random_normal with a third argument n, a random sample of size n will be simulated.
This is an implementation of the Box-Mueller algorithm, as described in Knuth, D.E. (1981) Seminumerical Algorithms. The Art of Computer Programming. Addison-Wesley.
To make use of this function, write first load("distrib").