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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").
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.
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
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").
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").
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").
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").
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").
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").
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").
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
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").
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").
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
Returns the standard deviation of a Student random variable t(n), with n>2. To make use of this function, write first load("distrib").
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").
Returns the kurtosis coefficient of a Student random variable t(n), with n>4. To make use of this function, write first load("distrib").
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").
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
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
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").
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
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").
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").
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").
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").
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").
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
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
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
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
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
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)
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)
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
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").
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").
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").
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.
Returns the mean of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Returns the variance of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Returns the standard deviation of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Returns the skewness coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Returns the kurtosis coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
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").
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").
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
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
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").
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").
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").
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").
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").
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").
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
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
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
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
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
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
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
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
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").
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").
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.
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
Returns the mean of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
Returns the variance of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
Returns the standard deviation of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
Returns the skewness coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
Returns the kurtosis coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").
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").
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").
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
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").
Returns the mean of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
Returns the variance of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
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").
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").
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").
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 combination 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").
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").
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
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").
Returns the mean of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
Returns the variance of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
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").
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").
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").
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").
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").
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").
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").
Returns the mean of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").
Returns the variance of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").
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").
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").
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").
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").
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").
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").
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").
Returns the mean of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
Returns the variance of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
Returns the standard deviation of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
Returns the skewness coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
Returns the kurtosis coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").
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").
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").
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").
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").
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").
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").
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").
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").
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").
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").
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").
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").
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").
Returns the mean of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
Returns the variance of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").
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").
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").
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").
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").
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
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
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
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
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
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
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
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
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").
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").
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").
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").
Returns the mean of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Returns the variance of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Returns the standard deviation of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Returns the skewness coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Returns the kurtosis coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
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").
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").
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").
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").
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").
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").
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").
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").
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.
Returns the variance of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
Returns the standard deviation of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
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.
Returns the kurtosis coefficient of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").
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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