(maxima.info)Functions and Variables for continuous distributions
52.2 Functions and Variables for continuous distributions
=========================================================
-- Function: pdf_normal (<x>,<m>,<s>)
Returns the value at <x> of the density function of a Normal(m,s)
random variable, with s>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_normal (<x>,<m>,<s>)
Returns the value at <x> of the distribution function of a
Normal(m,s) random variable, with s>0. This function is defined in
terms of Maxima's built-in error function 'erf'.
(%i1) load ("distrib")$
(%i2) cdf_normal(x,m,s);
x - m
erf(---------)
sqrt(2) s 1
(%o2) -------------- + -
2 2
See also 'erf'.
-- Function: quantile_normal (<q>,<m>,<s>)
Returns the <q>-quantile of a 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
-- Function: mean_normal (<m>,<s>)
Returns the mean of a Normal(m,s) random variable, with s>0, namely
<m>. To make use of this function, write first 'load("distrib")'.
-- Function: var_normal (<m>,<s>)
Returns the variance of a Normal(m,s) random variable, with s>0,
namely <s^2>. To make use of this function, write first
'load("distrib")'.
-- Function: std_normal (<m>,<s>)
Returns the standard deviation of a Normal(m,s) random variable,
with s>0, namely <s>. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_normal (<m>,<s>)
Returns the skewness coefficient of a Normal(m,s) random variable,
with s>0, which is always equal to 0. To make use of this
function, write first 'load("distrib")'.
-- Function: kurtosis_normal (<m>,<s>)
Returns the kurtosis coefficient of a Normal(m,s) random variable,
with s>0, which is always equal to 0. To make use of this
function, write first 'load("distrib")'.
-- Function: random_normal (<m>,<s>)
random_normal (<m>,<s>,<n>)
Returns a 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")'.
-- Function: pdf_student_t (<x>,<n>)
Returns the value at <x> of the density function of a Student
random variable t(n), with n>0 degrees of freedom. To make use of
this function, write first 'load("distrib")'.
-- Function: cdf_student_t (<x>,<n>)
Returns the value at <x> of the distribution function of a Student
random variable t(n), with n>0 degrees of freedom.
(%i1) load ("distrib")$
(%i2) cdf_student_t(1/2, 7/3);
7 1 28
beta_incomplete_regularized(-, -, --)
6 2 31
(%o2) 1 - -------------------------------------
2
(%i3) float(%);
(%o3) .6698450596140415
-- Function: quantile_student_t (<q>,<n>)
Returns the <q>-quantile of a Student random variable t(n), with
n>0; in other words, this is the inverse of 'cdf_student_t'.
Argument <q> must be an element of [0,1]. To make use of this
function, write first 'load("distrib")'.
-- Function: mean_student_t (<n>)
Returns the mean of a Student random variable t(n), with n>0, which
is always equal to 0. To make use of this function, write first
'load("distrib")'.
-- Function: var_student_t (<n>)
Returns the variance of a Student random variable t(n), with n>2.
(%i1) load ("distrib")$
(%i2) var_student_t(n);
n
(%o2) -----
n - 2
-- Function: std_student_t (<n>)
Returns the standard deviation of a Student random variable t(n),
with n>2. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_student_t (<n>)
Returns the skewness coefficient of a Student random variable t(n),
with n>3, which is always equal to 0. To make use of this
function, write first 'load("distrib")'.
-- Function: kurtosis_student_t (<n>)
Returns the kurtosis coefficient of a Student random variable t(n),
with n>4. To make use of this function, write first
'load("distrib")'.
-- Function: random_student_t (<n>)
random_student_t (<n>,<m>)
Returns a Student random variate t(n), with n>0. Calling
'random_student_t' with a second argument <m>, a random sample of
size <m> will be simulated.
The implemented algorithm is based on the fact that if <Z> is a
normal random variable N(0,1) and S^2 is a chi square random
variable with <n> degrees of freedom, Chi^2(n), then
Z
X = -------------
/ 2 \ 1/2
| S |
| --- |
\ n /
is a Student random variable with <n> degrees of freedom, t(n).
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_noncentral_student_t (<x>,<n>,<ncp>)
Returns the value at <x> of the density function of a noncentral
Student random variable nc_t(n,ncp), with n>0 degrees of freedom
and noncentrality parameter ncp. To make use of this function,
write first 'load("distrib")'.
Sometimes an extra work is necessary to get the final result.
(%i1) load ("distrib")$
(%i2) expand(pdf_noncentral_student_t(3,5,0.1));
7/2 7/2
0.04296414417400905 5 1.323650307289301e-6 5
(%o2) ------------------------ + -------------------------
3/2 5/2 sqrt(%pi)
2 14 sqrt(%pi)
7/2
1.94793720435093e-4 5
+ ------------------------
%pi
(%i3) float(%);
(%o3) .02080593159405669
-- Function: cdf_noncentral_student_t (<x>,<n>,<ncp>)
Returns the value at <x> of the distribution function of a
noncentral Student random variable nc_t(n,ncp), with n>0 degrees of
freedom and noncentrality parameter ncp. This function has no
closed form and it is numerically computed.
(%i1) load ("distrib")$
(%i2) cdf_noncentral_student_t(-2,5,-5);
(%o2) .9952030093319743
-- Function: quantile_noncentral_student_t (<q>,<n>,<ncp>)
Returns the <q>-quantile of a noncentral Student random variable
nc_t(n,ncp), with n>0 degrees of freedom and noncentrality
parameter ncp; in other words, this is the inverse of
'cdf_noncentral_student_t'. Argument <q> must be an element of
[0,1]. To make use of this function, write first
'load("distrib")'.
-- Function: mean_noncentral_student_t (<n>,<ncp>)
Returns the mean of a noncentral Student random variable
nc_t(n,ncp), with n>1 degrees of freedom and noncentrality
parameter ncp. To make use of this function, write first
'load("distrib")'.
(%i1) load ("distrib")$
(%i2) mean_noncentral_student_t(df,k);
df - 1
gamma(------) sqrt(df) k
2
(%o2) ------------------------
df
sqrt(2) gamma(--)
2
-- Function: var_noncentral_student_t (<n>,<ncp>)
Returns the variance of a noncentral Student random variable
nc_t(n,ncp), with n>2 degrees of freedom and noncentrality
parameter ncp. To make use of this function, write first
'load("distrib")'.
-- Function: std_noncentral_student_t (<n>,<ncp>)
Returns the standard deviation of a noncentral Student random
variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality
parameter ncp. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_noncentral_student_t (<n>,<ncp>)
Returns the skewness coefficient of a noncentral Student random
variable nc_t(n,ncp), with n>3 degrees of freedom and noncentrality
parameter ncp. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_noncentral_student_t (<n>,<ncp>)
Returns the kurtosis coefficient of a noncentral Student random
variable nc_t(n,ncp), with n>4 degrees of freedom and noncentrality
parameter ncp. To make use of this function, write first
'load("distrib")'.
-- Function: random_noncentral_student_t (<n>,<ncp>)
random_noncentral_student_t (<n>,<ncp>,<m>)
Returns a noncentral Student random variate nc_t(n,ncp), with n>0.
Calling 'random_noncentral_student_t' with a third argument <m>, a
random sample of size <m> will be simulated.
The implemented algorithm is based on the fact that if <X> is a
normal random variable N(ncp,1) and S^2 is a chi square random
variable with <n> degrees of freedom, Chi^2(n), then
X
U = -------------
/ 2 \ 1/2
| S |
| --- |
\ n /
is a noncentral Student random variable with <n> degrees of freedom
and noncentrality parameter ncp, nc_t(n,ncp).
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_chi2 (<x>,<n>)
Returns the value at <x> of the density function of a Chi-square
random variable Chi^2(n), with n>0. The Chi^2(n) random variable
is equivalent to the Gamma(n/2,2).
(%i1) load ("distrib")$
(%i2) pdf_chi2(x,n);
n/2 - 1 - x/2
x %e
(%o2) ----------------
n/2 n
2 gamma(-)
2
-- Function: cdf_chi2 (<x>,<n>)
Returns the value at <x> of the distribution function of a
Chi-square random variable Chi^2(n), with n>0.
(%i1) load ("distrib")$
(%i2) cdf_chi2(3,4);
3
(%o2) 1 - gamma_incomplete_regularized(2, -)
2
(%i3) float(%);
(%o3) .4421745996289256
-- Function: quantile_chi2 (<q>,<n>)
Returns the <q>-quantile of a Chi-square random variable Chi^2(n),
with n>0; in other words, this is the inverse of 'cdf_chi2'.
Argument <q> must be an element of [0,1].
This function has no closed form and it is numerically computed.
(%i1) load ("distrib")$
(%i2) quantile_chi2(0.99,9);
(%o2) 21.66599433346194
-- Function: mean_chi2 (<n>)
Returns the mean of a Chi-square random variable Chi^2(n), with
n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).
(%i1) load ("distrib")$
(%i2) mean_chi2(n);
(%o2) n
-- Function: var_chi2 (<n>)
Returns the variance of a Chi-square random variable Chi^2(n), with
n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).
(%i1) load ("distrib")$
(%i2) var_chi2(n);
(%o2) 2 n
-- Function: std_chi2 (<n>)
Returns the standard deviation of a Chi-square random variable
Chi^2(n), with n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).
(%i1) load ("distrib")$
(%i2) std_chi2(n);
(%o2) sqrt(2) sqrt(n)
-- Function: skewness_chi2 (<n>)
Returns the skewness coefficient of a Chi-square random variable
Chi^2(n), with n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).
(%i1) load ("distrib")$
(%i2) skewness_chi2(n);
3/2
2
(%o2) -------
sqrt(n)
-- Function: kurtosis_chi2 (<n>)
Returns the kurtosis coefficient of a Chi-square random variable
Chi^2(n), with n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).
(%i1) load ("distrib")$
(%i2) kurtosis_chi2(n);
12
(%o2) --
n
-- Function: random_chi2 (<n>)
random_chi2 (<n>,<m>)
Returns a Chi-square random variate Chi^2(n), with n>0. Calling
'random_chi2' with a second argument <m>, a random sample of size
<m> will be simulated.
The simulation is based on the Ahrens-Cheng algorithm. See
'random_gamma' for details.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_noncentral_chi2 (<x>,<n>,<ncp>)
Returns the value at <x> of the density function of a noncentral
Chi-square random variable nc_Chi^2(n,ncp), with n>0 and
noncentrality parameter ncp>=0. To make use of this function,
write first 'load("distrib")'.
-- Function: cdf_noncentral_chi2 (<x>,<n>,<ncp>)
Returns the value at <x> of the distribution function of a
noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and
noncentrality parameter ncp>=0. To make use of this function,
write first 'load("distrib")'.
-- Function: quantile_noncentral_chi2 (<q>,<n>,<ncp>)
Returns the <q>-quantile of a noncentral Chi-square random variable
nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0; in
other words, this is the inverse of 'cdf_noncentral_chi2'.
Argument <q> must be an element of [0,1].
This function has no closed form and it is numerically computed.
-- Function: mean_noncentral_chi2 (<n>,<ncp>)
Returns the mean of a noncentral Chi-square random variable
nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
-- Function: var_noncentral_chi2 (<n>,<ncp>)
Returns the variance of a noncentral Chi-square random variable
nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
-- Function: std_noncentral_chi2 (<n>,<ncp>)
Returns the standard deviation of a noncentral Chi-square random
variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
ncp>=0.
-- Function: skewness_noncentral_chi2 (<n>,<ncp>)
Returns the skewness coefficient of a noncentral Chi-square random
variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
ncp>=0.
-- Function: kurtosis_noncentral_chi2 (<n>,<ncp>)
Returns the kurtosis coefficient of a noncentral Chi-square random
variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
ncp>=0.
-- Function: random_noncentral_chi2 (<n>,<ncp>)
random_noncentral_chi2 (<n>,<ncp>,<m>)
Returns a noncentral Chi-square random variate nc_Chi^2(n,ncp),
with n>0 and noncentrality parameter ncp>=0. Calling
'random_noncentral_chi2' with a third argument <m>, a random sample
of size <m> will be simulated.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_f (<x>,<m>,<n>)
Returns the value at <x> of the density function of a F random
variable F(m,n), with m,n>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_f (<x>,<m>,<n>)
Returns the value at <x> of the distribution function of a F random
variable F(m,n), with m,n>0.
(%i1) load ("distrib")$
(%i2) cdf_f(2,3,9/4);
9 3 3
(%o2) 1 - beta_incomplete_regularized(-, -, --)
8 2 11
(%i3) float(%);
(%o3) 0.66756728179008
-- Function: quantile_f (<q>,<m>,<n>)
Returns the <q>-quantile of a F random variable F(m,n), with m,n>0;
in other words, this is the inverse of 'cdf_f'. Argument <q> must
be an element of [0,1].
(%i1) load ("distrib")$
(%i2) quantile_f(2/5,sqrt(3),5);
(%o2) 0.518947838573693
-- Function: mean_f (<m>,<n>)
Returns the mean of a F random variable F(m,n), with m>0, n>2. To
make use of this function, write first 'load("distrib")'.
-- Function: var_f (<m>,<n>)
Returns the variance of a F random variable F(m,n), with m>0, n>4.
To make use of this function, write first 'load("distrib")'.
-- Function: std_f (<m>,<n>)
Returns the standard deviation of a F random variable F(m,n), with
m>0, n>4. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_f (<m>,<n>)
Returns the skewness coefficient of a F random variable F(m,n),
with m>0, n>6. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_f (<m>,<n>)
Returns the kurtosis coefficient of a F random variable F(m,n),
with m>0, n>8. To make use of this function, write first
'load("distrib")'.
-- Function: random_f (<m>,<n>)
random_f (<m>,<n>,<k>)
Returns a F random variate F(m,n), with m,n>0. Calling 'random_f'
with a third argument <k>, a random sample of size <k> will be
simulated.
The simulation algorithm is based on the fact that if <X> is a
Chi^2(m) random variable and Y is a Chi^2(n) random variable, then
n X
F = ---
m Y
is a F random variable with <m> and <n> degrees of freedom, F(m,n).
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_exp (<x>,<m>)
Returns the value at <x> of the density function of an
Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the
Weibull(1,1/m).
(%i1) load ("distrib")$
(%i2) pdf_exp(x,m);
- m x
(%o2) m %e
-- Function: cdf_exp (<x>,<m>)
Returns the value at <x> of the distribution function of an
Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the
Weibull(1,1/m).
(%i1) load ("distrib")$
(%i2) cdf_exp(x,m);
- m x
(%o2) 1 - %e
-- Function: quantile_exp (<q>,<m>)
Returns the <q>-quantile of an Exponential(m) random variable, with
m>0; in other words, this is the inverse of 'cdf_exp'. Argument
<q> must be an element of [0,1].
The Exponential(m) random variable is equivalent to the
Weibull(1,1/m).
(%i1) load ("distrib")$
(%i2) quantile_exp(0.56,5);
(%o2) .1641961104139661
(%i3) quantile_exp(0.56,m);
0.8209805520698303
(%o3) ------------------
m
-- Function: mean_exp (<m>)
Returns the mean of an Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the
Weibull(1,1/m).
(%i1) load ("distrib")$
(%i2) mean_exp(m);
1
(%o2) -
m
-- Function: var_exp (<m>)
Returns the variance of an Exponential(m) random variable, with
m>0.
The Exponential(m) random variable is equivalent to the
Weibull(1,1/m).
(%i1) load ("distrib")$
(%i2) var_exp(m);
1
(%o2) --
2
m
-- Function: std_exp (<m>)
Returns the standard deviation of an Exponential(m) random
variable, with m>0.
The Exponential(m) random variable is equivalent to the
Weibull(1,1/m).
(%i1) load ("distrib")$
(%i2) std_exp(m);
1
(%o2) -
m
-- Function: skewness_exp (<m>)
Returns the skewness coefficient of an Exponential(m) random
variable, with m>0.
The Exponential(m) random variable is equivalent to the
Weibull(1,1/m).
(%i1) load ("distrib")$
(%i2) skewness_exp(m);
(%o2) 2
-- Function: kurtosis_exp (<m>)
Returns the kurtosis coefficient of an Exponential(m) random
variable, with m>0.
The Exponential(m) random variable is equivalent to the
Weibull(1,1/m).
(%i1) load ("distrib")$
(%i2) kurtosis_exp(m);
(%o3) 6
-- Function: random_exp (<m>)
random_exp (<m>,<k>)
Returns an Exponential(m) random variate, with m>0. Calling
'random_exp' with a second argument <k>, a random sample of size
<k> will be simulated.
The simulation algorithm is based on the general inverse method.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_lognormal (<x>,<m>,<s>)
Returns the value at <x> of the density function of a
Lognormal(m,s) random variable, with s>0. To make use of this
function, write first 'load("distrib")'.
-- Function: cdf_lognormal (<x>,<m>,<s>)
Returns the value at <x> of the distribution function of a
Lognormal(m,s) random variable, with s>0. This function is defined
in terms of Maxima's built-in error function 'erf'.
(%i1) load ("distrib")$
(%i2) cdf_lognormal(x,m,s);
log(x) - m
erf(----------)
sqrt(2) s 1
(%o2) --------------- + -
2 2
See also 'erf'.
-- Function: quantile_lognormal (<q>,<m>,<s>)
Returns the <q>-quantile of a 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
-- Function: mean_lognormal (<m>,<s>)
Returns the mean of a Lognormal(m,s) random variable, with s>0. To
make use of this function, write first 'load("distrib")'.
-- Function: var_lognormal (<m>,<s>)
Returns the variance of a Lognormal(m,s) random variable, with s>0.
To make use of this function, write first 'load("distrib")'.
-- Function: std_lognormal (<m>,<s>)
Returns the standard deviation of a Lognormal(m,s) random variable,
with s>0. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_lognormal (<m>,<s>)
Returns the skewness coefficient of a Lognormal(m,s) random
variable, with s>0. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_lognormal (<m>,<s>)
Returns the kurtosis coefficient of a Lognormal(m,s) random
variable, with s>0. To make use of this function, write first
'load("distrib")'.
-- Function: random_lognormal (<m>,<s>)
random_lognormal (<m>,<s>,<n>)
Returns a 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")'.
-- Function: pdf_gamma (<x>,<a>,<b>)
Returns the value at <x> of the density function of a Gamma(a,b)
random variable, with a,b>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_gamma (<x>,<a>,<b>)
Returns the value at <x> of the distribution function of a
Gamma(a,b) random variable, with a,b>0.
(%i1) load ("distrib")$
(%i2) cdf_gamma(3,5,21);
1
(%o2) 1 - gamma_incomplete_regularized(5, -)
7
(%i3) float(%);
(%o3) 4.402663157376807E-7
-- Function: quantile_gamma (<q>,<a>,<b>)
Returns the <q>-quantile of a Gamma(a,b) random variable, with
a,b>0; in other words, this is the inverse of 'cdf_gamma'.
Argument <q> must be an element of [0,1]. To make use of this
function, write first 'load("distrib")'.
-- Function: mean_gamma (<a>,<b>)
Returns the mean of a Gamma(a,b) random variable, with a,b>0. To
make use of this function, write first 'load("distrib")'.
-- Function: var_gamma (<a>,<b>)
Returns the variance of a Gamma(a,b) random variable, with a,b>0.
To make use of this function, write first 'load("distrib")'.
-- Function: std_gamma (<a>,<b>)
Returns the standard deviation of a Gamma(a,b) random variable,
with a,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_gamma (<a>,<b>)
Returns the skewness coefficient of a Gamma(a,b) random variable,
with a,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_gamma (<a>,<b>)
Returns the kurtosis coefficient of a Gamma(a,b) random variable,
with a,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: random_gamma (<a>,<b>)
random_gamma (<a>,<b>,<n>)
Returns a Gamma(a,b) random variate, with a,b>0. Calling
'random_gamma' with a third argument <n>, a random sample of size
<n> will be simulated.
The implemented algorithm is a combinantion of two procedures,
depending on the value of parameter <a>:
For a>=1, Cheng, R.C.H. and Feast, G.M. (1979). <Some simple gamma
variate generators>. Appl. Stat., 28, 3, 290-295.
For 0<a<1, Ahrens, J.H. and Dieter, U. (1974). <Computer methods
for sampling from gamma, beta, poisson and binomial
cdf_tributions>. Computing, 12, 223-246.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_beta (<x>,<a>,<b>)
Returns the value at <x> of the density function of a Beta(a,b)
random variable, with a,b>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_beta (<x>,<a>,<b>)
Returns the value at <x> of the distribution function of a
Beta(a,b) random variable, with a,b>0.
(%i1) load ("distrib")$
(%i2) cdf_beta(1/3,15,2);
11
(%o2) --------
14348907
(%i3) float(%);
(%o3) 7.666089131388195E-7
-- Function: quantile_beta (<q>,<a>,<b>)
Returns the <q>-quantile of a Beta(a,b) random variable, with
a,b>0; in other words, this is the inverse of 'cdf_beta'. Argument
<q> must be an element of [0,1]. To make use of this function,
write first 'load("distrib")'.
-- Function: mean_beta (<a>,<b>)
Returns the mean of a Beta(a,b) random variable, with a,b>0. To
make use of this function, write first 'load("distrib")'.
-- Function: var_beta (<a>,<b>)
Returns the variance of a Beta(a,b) random variable, with a,b>0.
To make use of this function, write first 'load("distrib")'.
-- Function: std_beta (<a>,<b>)
Returns the standard deviation of a Beta(a,b) random variable, with
a,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_beta (<a>,<b>)
Returns the skewness coefficient of a Beta(a,b) random variable,
with a,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_beta (<a>,<b>)
Returns the kurtosis coefficient of a Beta(a,b) random variable,
with a,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: random_beta (<a>,<b>)
random_beta (<a>,<b>,<n>)
Returns a Beta(a,b) random variate, with a,b>0. Calling
'random_beta' with a third argument <n>, a random sample of size
<n> will be simulated.
The implemented algorithm is defined in Cheng, R.C.H. (1978).
<Generating Beta Variates with Nonintegral Shape Parameters>.
Communications of the ACM, 21:317-322
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_continuous_uniform (<x>,<a>,<b>)
Returns the value at <x> of the density function of a Continuous
Uniform(a,b) random variable, with a<b. To make use of this
function, write first 'load("distrib")'.
-- Function: cdf_continuous_uniform (<x>,<a>,<b>)
Returns the value at <x> of the distribution function of a
Continuous Uniform(a,b) random variable, with a<b. To make use of
this function, write first 'load("distrib")'.
-- Function: quantile_continuous_uniform (<q>,<a>,<b>)
Returns the <q>-quantile of a Continuous Uniform(a,b) random
variable, with a<b; in other words, this is the inverse of
'cdf_continuous_uniform'. Argument <q> must be an element of
[0,1]. To make use of this function, write first
'load("distrib")'.
-- Function: mean_continuous_uniform (<a>,<b>)
Returns the mean of a Continuous Uniform(a,b) random variable, with
a<b. To make use of this function, write first 'load("distrib")'.
-- Function: var_continuous_uniform (<a>,<b>)
Returns the variance of a Continuous Uniform(a,b) random variable,
with a<b. To make use of this function, write first
'load("distrib")'.
-- Function: std_continuous_uniform (<a>,<b>)
Returns the standard deviation of a Continuous Uniform(a,b) random
variable, with a<b. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_continuous_uniform (<a>,<b>)
Returns the skewness coefficient of a Continuous Uniform(a,b)
random variable, with a<b. To make use of this function, write
first 'load("distrib")'.
-- Function: kurtosis_continuous_uniform (<a>,<b>)
Returns the kurtosis coefficient of a Continuous Uniform(a,b)
random variable, with a<b. To make use of this function, write
first 'load("distrib")'.
-- Function: random_continuous_uniform (<a>,<b>)
random_continuous_uniform (<a>,<b>,<n>)
Returns a Continuous Uniform(a,b) random variate, with a<b.
Calling 'random_continuous_uniform' with a third argument <n>, a
random sample of size <n> will be simulated.
This is a direct application of the 'random' built-in Maxima
function.
See also 'random'. To make use of this function, write first
'load("distrib")'.
-- Function: pdf_logistic (<x>,<a>,<b>)
Returns the value at <x> of the density function of a Logistic(a,b)
random variable , with b>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_logistic (<x>,<a>,<b>)
Returns the value at <x> of the distribution function of a
Logistic(a,b) random variable , with b>0. To make use of this
function, write first 'load("distrib")'.
-- Function: quantile_logistic (<q>,<a>,<b>)
Returns the <q>-quantile of a Logistic(a,b) random variable , with
b>0; in other words, this is the inverse of 'cdf_logistic'.
Argument <q> must be an element of [0,1]. To make use of this
function, write first 'load("distrib")'.
-- Function: mean_logistic (<a>,<b>)
Returns the mean of a Logistic(a,b) random variable , with b>0. To
make use of this function, write first 'load("distrib")'.
-- Function: var_logistic (<a>,<b>)
Returns the variance of a Logistic(a,b) random variable , with b>0.
To make use of this function, write first 'load("distrib")'.
-- Function: std_logistic (<a>,<b>)
Returns the standard deviation of a Logistic(a,b) random variable ,
with b>0. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_logistic (<a>,<b>)
Returns the skewness coefficient of a Logistic(a,b) random variable
, with b>0. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_logistic (<a>,<b>)
Returns the kurtosis coefficient of a Logistic(a,b) random variable
, with b>0. To make use of this function, write first
'load("distrib")'.
-- Function: random_logistic (<a>,<b>)
random_logistic (<a>,<b>,<n>)
Returns a Logistic(a,b) random variate, with b>0. Calling
'random_logistic' with a third argument <n>, a random sample of
size <n> will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_pareto (<x>,<a>,<b>)
Returns the value at <x> of the density function of a Pareto(a,b)
random variable, with a,b>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_pareto (<x>,<a>,<b>)
Returns the value at <x> of the distribution function of a
Pareto(a,b) random variable, with a,b>0. To make use of this
function, write first 'load("distrib")'.
-- Function: quantile_pareto (<q>,<a>,<b>)
Returns the <q>-quantile of a Pareto(a,b) random variable, with
a,b>0; in other words, this is the inverse of 'cdf_pareto'.
Argument <q> must be an element of [0,1]. To make use of this
function, write first 'load("distrib")'.
-- Function: mean_pareto (<a>,<b>)
Returns the mean of a Pareto(a,b) random variable, with a>1,b>0.
To make use of this function, write first 'load("distrib")'.
-- Function: var_pareto (<a>,<b>)
Returns the variance of a Pareto(a,b) random variable, with
a>2,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: std_pareto (<a>,<b>)
Returns the standard deviation of a Pareto(a,b) random variable,
with a>2,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_pareto (<a>,<b>)
Returns the skewness coefficient of a Pareto(a,b) random variable,
with a>3,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_pareto (<a>,<b>)
Returns the kurtosis coefficient of a Pareto(a,b) random variable,
with a>4,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: random_pareto (<a>,<b>)
random_pareto (<a>,<b>,<n>)
Returns a Pareto(a,b) random variate, with a>0,b>0. Calling
'random_pareto' with a third argument <n>, a random sample of size
<n> will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_weibull (<x>,<a>,<b>)
Returns the value at <x> of the density function of a Weibull(a,b)
random variable, with a,b>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_weibull (<x>,<a>,<b>)
Returns the value at <x> of the distribution function of a
Weibull(a,b) random variable, with a,b>0. To make use of this
function, write first 'load("distrib")'.
-- Function: quantile_weibull (<q>,<a>,<b>)
Returns the <q>-quantile of a Weibull(a,b) random variable, with
a,b>0; in other words, this is the inverse of 'cdf_weibull'.
Argument <q> must be an element of [0,1]. To make use of this
function, write first 'load("distrib")'.
-- Function: mean_weibull (<a>,<b>)
Returns the mean of a Weibull(a,b) random variable, with a,b>0. To
make use of this function, write first 'load("distrib")'.
-- Function: var_weibull (<a>,<b>)
Returns the variance of a Weibull(a,b) random variable, with a,b>0.
To make use of this function, write first 'load("distrib")'.
-- Function: std_weibull (<a>,<b>)
Returns the standard deviation of a Weibull(a,b) random variable,
with a,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_weibull (<a>,<b>)
Returns the skewness coefficient of a Weibull(a,b) random variable,
with a,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_weibull (<a>,<b>)
Returns the kurtosis coefficient of a Weibull(a,b) random variable,
with a,b>0. To make use of this function, write first
'load("distrib")'.
-- Function: random_weibull (<a>,<b>)
random_weibull (<a>,<b>,<n>)
Returns a Weibull(a,b) random variate, with a,b>0. Calling
'random_weibull' with a third argument <n>, a random sample of size
<n> will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_rayleigh (<x>,<b>)
Returns the value at <x> of the density function of a Rayleigh(b)
random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the
Weibull(2,1/b).
(%i1) load ("distrib")$
(%i2) pdf_rayleigh(x,b);
2 2
2 - b x
(%o2) 2 b x %e
-- Function: cdf_rayleigh (<x>,<b>)
Returns the value at <x> of the distribution function of a
Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the
Weibull(2,1/b).
(%i1) load ("distrib")$
(%i2) cdf_rayleigh(x,b);
2 2
- b x
(%o2) 1 - %e
-- Function: quantile_rayleigh (<q>,<b>)
Returns the <q>-quantile of a Rayleigh(b) random variable, with
b>0; in other words, this is the inverse of 'cdf_rayleigh'.
Argument <q> must be an element of [0,1].
The Rayleigh(b) random variable is equivalent to the
Weibull(2,1/b).
(%i1) load ("distrib")$
(%i2) quantile_rayleigh(0.99,b);
2.145966026289347
(%o2) -----------------
b
-- Function: mean_rayleigh (<b>)
Returns the mean of a Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the
Weibull(2,1/b).
(%i1) load ("distrib")$
(%i2) mean_rayleigh(b);
sqrt(%pi)
(%o2) ---------
2 b
-- Function: var_rayleigh (<b>)
Returns the variance of a Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the
Weibull(2,1/b).
(%i1) load ("distrib")$
(%i2) var_rayleigh(b);
%pi
1 - ---
4
(%o2) -------
2
b
-- Function: std_rayleigh (<b>)
Returns the standard deviation of a Rayleigh(b) random variable,
with b>0.
The Rayleigh(b) random variable is equivalent to the
Weibull(2,1/b).
(%i1) load ("distrib")$
(%i2) std_rayleigh(b);
%pi
sqrt(1 - ---)
4
(%o2) -------------
b
-- Function: skewness_rayleigh (<b>)
Returns the skewness coefficient of a Rayleigh(b) random variable,
with b>0.
The Rayleigh(b) random variable is equivalent to the
Weibull(2,1/b).
(%i1) load ("distrib")$
(%i2) skewness_rayleigh(b);
3/2
%pi 3 sqrt(%pi)
------ - -----------
4 4
(%o2) --------------------
%pi 3/2
(1 - ---)
4
-- Function: kurtosis_rayleigh (<b>)
Returns the kurtosis coefficient of a Rayleigh(b) random variable,
with b>0.
The Rayleigh(b) random variable is equivalent to the
Weibull(2,1/b).
(%i1) load ("distrib")$
(%i2) kurtosis_rayleigh(b);
2
3 %pi
2 - ------
16
(%o2) ---------- - 3
%pi 2
(1 - ---)
4
-- Function: random_rayleigh (<b>)
random_rayleigh (<b>,<n>)
Returns a Rayleigh(b) random variate, with b>0. Calling
'random_rayleigh' with a second argument <n>, a random sample of
size <n> will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_laplace (<x>,<a>,<b>)
Returns the value at <x> of the density function of a Laplace(a,b)
random variable, with b>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_laplace (<x>,<a>,<b>)
Returns the value at <x> of the distribution function of a
Laplace(a,b) random variable, with b>0. To make use of this
function, write first 'load("distrib")'.
-- Function: quantile_laplace (<q>,<a>,<b>)
Returns the <q>-quantile of a Laplace(a,b) random variable, with
b>0; in other words, this is the inverse of 'cdf_laplace'.
Argument <q> must be an element of [0,1]. To make use of this
function, write first 'load("distrib")'.
-- Function: mean_laplace (<a>,<b>)
Returns the mean of a Laplace(a,b) random variable, with b>0. To
make use of this function, write first 'load("distrib")'.
-- Function: var_laplace (<a>,<b>)
Returns the variance of a Laplace(a,b) random variable, with b>0.
To make use of this function, write first 'load("distrib")'.
-- Function: std_laplace (<a>,<b>)
Returns the standard deviation of a Laplace(a,b) random variable,
with b>0. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_laplace (<a>,<b>)
Returns the skewness coefficient of a Laplace(a,b) random variable,
with b>0. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_laplace (<a>,<b>)
Returns the kurtosis coefficient of a Laplace(a,b) random variable,
with b>0. To make use of this function, write first
'load("distrib")'.
-- Function: random_laplace (<a>,<b>)
random_laplace (<a>,<b>,<n>)
Returns a Laplace(a,b) random variate, with b>0. Calling
'random_laplace' with a third argument <n>, a random sample of size
<n> will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_cauchy (<x>,<a>,<b>)
Returns the value at <x> of the density function of a Cauchy(a,b)
random variable, with b>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_cauchy (<x>,<a>,<b>)
Returns the value at <x> of the distribution function of a
Cauchy(a,b) random variable, with b>0. To make use of this
function, write first 'load("distrib")'.
-- Function: quantile_cauchy (<q>,<a>,<b>)
Returns the <q>-quantile of a Cauchy(a,b) random variable, with
b>0; in other words, this is the inverse of 'cdf_cauchy'. Argument
<q> must be an element of [0,1]. To make use of this function,
write first 'load("distrib")'.
-- Function: random_cauchy (<a>,<b>)
random_cauchy (<a>,<b>,<n>)
Returns a Cauchy(a,b) random variate, with b>0. Calling
'random_cauchy' with a third argument <n>, a random sample of size
<n> will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_gumbel (<x>,<a>,<b>)
Returns the value at <x> of the density function of a Gumbel(a,b)
random variable, with b>0. To make use of this function, write
first 'load("distrib")'.
-- Function: cdf_gumbel (<x>,<a>,<b>)
Returns the value at <x> of the distribution function of a
Gumbel(a,b) random variable, with b>0. To make use of this
function, write first 'load("distrib")'.
-- Function: quantile_gumbel (<q>,<a>,<b>)
Returns the <q>-quantile of a Gumbel(a,b) random variable, with
b>0; in other words, this is the inverse of 'cdf_gumbel'. Argument
<q> must be an element of [0,1]. To make use of this function,
write first 'load("distrib")'.
-- Function: mean_gumbel (<a>,<b>)
Returns the mean of a Gumbel(a,b) random variable, with b>0.
(%i1) load ("distrib")$
(%i2) mean_gumbel(a,b);
(%o2) %gamma b + a
where symbol '%gamma' stands for the Euler-Mascheroni constant.
See also '%gamma'.
-- Function: var_gumbel (<a>,<b>)
Returns the variance of a Gumbel(a,b) random variable, with b>0.
To make use of this function, write first 'load("distrib")'.
-- Function: std_gumbel (<a>,<b>)
Returns the standard deviation of a Gumbel(a,b) random variable,
with b>0. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_gumbel (<a>,<b>)
Returns the skewness coefficient of a Gumbel(a,b) random variable,
with b>0.
(%i1) load ("distrib")$
(%i2) skewness_gumbel(a,b);
3/2
2 6 zeta(3)
(%o2) --------------
3
%pi
where 'zeta' stands for the Riemann's zeta function.
-- Function: kurtosis_gumbel (<a>,<b>)
Returns the kurtosis coefficient of a Gumbel(a,b) random variable,
with b>0. To make use of this function, write first
'load("distrib")'.
-- Function: random_gumbel (<a>,<b>)
random_gumbel (<a>,<b>,<n>)
Returns a Gumbel(a,b) random variate, with b>0. Calling
'random_gumbel' with a third argument <n>, a random sample of size
<n> will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first 'load("distrib")'.
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