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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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