(maxima.info)Functions and Variables for discrete distributions
52.3 Functions and Variables for discrete distributions
=======================================================
-- Function: pdf_general_finite_discrete (<x>,<v>)
Returns the value at <x> of the probability function of a general
finite discrete random variable, with vector probabilities v, such
that 'Pr(X=i) = v_i'. Vector v can be a list of nonnegative
expressions, whose components will be normalized to get a vector of
probabilities. To make use of this function, write first
'load("distrib")'.
(%i1) load ("distrib")$
(%i2) pdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
4
(%o2) -
7
(%i3) pdf_general_finite_discrete(2, [1, 4, 2]);
4
(%o3) -
7
-- Function: cdf_general_finite_discrete (<x>,<v>)
Returns the value at <x> of the distribution function of a general
finite discrete random variable, with vector probabilities v.
See 'pdf_general_finite_discrete' for more details.
(%i1) load ("distrib")$
(%i2) cdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
5
(%o2) -
7
(%i3) cdf_general_finite_discrete(2, [1, 4, 2]);
5
(%o3) -
7
(%i4) cdf_general_finite_discrete(2+1/2, [1, 4, 2]);
5
(%o4) -
7
-- Function: quantile_general_finite_discrete (<q>,<v>)
Returns the <q>-quantile of a general finite discrete random
variable, with vector probabilities v.
See 'pdf_general_finite_discrete' for more details.
-- Function: mean_general_finite_discrete (<v>)
Returns the mean of a general finite discrete random variable, with
vector probabilities v.
See 'pdf_general_finite_discrete' for more details.
-- Function: var_general_finite_discrete (<v>)
Returns the variance of a general finite discrete random variable,
with vector probabilities v.
See 'pdf_general_finite_discrete' for more details.
-- Function: std_general_finite_discrete (<v>)
Returns the standard deviation of a general finite discrete random
variable, with vector probabilities v.
See 'pdf_general_finite_discrete' for more details.
-- Function: skewness_general_finite_discrete (<v>)
Returns the skewness coefficient of a general finite discrete
random variable, with vector probabilities v.
See 'pdf_general_finite_discrete' for more details.
-- Function: kurtosis_general_finite_discrete (<v>)
Returns the kurtosis coefficient of a general finite discrete
random variable, with vector probabilities v.
See 'pdf_general_finite_discrete' for more details.
-- Function: random_general_finite_discrete (<v>)
random_general_finite_discrete (<v>,<m>)
Returns a general finite discrete random variate, with vector
probabilities v. Calling 'random_general_finite_discrete' with a
second argument <m>, a random sample of size <m> will be simulated.
See 'pdf_general_finite_discrete' for more details.
(%i1) load ("distrib")$
(%i2) random_general_finite_discrete([1,3,1,5]);
(%o2) 4
(%i3) random_general_finite_discrete([1,3,1,5], 10);
(%o3) [4, 2, 2, 3, 2, 4, 4, 1, 2, 2]
-- Function: pdf_binomial (<x>,<n>,<p>)
Returns the value at <x> of the probability function of a
Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a
positive integer. To make use of this function, write first
'load("distrib")'.
-- Function: cdf_binomial (<x>,<n>,<p>)
Returns the value at <x> of the distribution function of a
Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a
positive integer.
(%i1) load ("distrib")$
(%i2) cdf_binomial(5,7,1/6);
7775
(%o2) ----
7776
(%i3) float(%);
(%o3) .9998713991769548
-- Function: quantile_binomial (<q>,<n>,<p>)
Returns the <q>-quantile of a Binomial(n,p) random variable, with 0
\leq p \leq 1 and n a positive integer; in other words, this is the
inverse of 'cdf_binomial'. Argument <q> must be an element of
[0,1]. To make use of this function, write first
'load("distrib")'.
-- Function: mean_binomial (<n>,<p>)
Returns the mean of a Binomial(n,p) random variable, with 0 \leq p
\leq 1 and n a positive integer. To make use of this function,
write first 'load("distrib")'.
-- Function: var_binomial (<n>,<p>)
Returns the variance of a Binomial(n,p) random variable, with 0
\leq p \leq 1 and n a positive integer. To make use of this
function, write first 'load("distrib")'.
-- Function: std_binomial (<n>,<p>)
Returns the standard deviation of a Binomial(n,p) random variable,
with 0 \leq p \leq 1 and n a positive integer. To make use of this
function, write first 'load("distrib")'.
-- Function: skewness_binomial (<n>,<p>)
Returns the skewness coefficient of a Binomial(n,p) random
variable, with 0 \leq p \leq 1 and n a positive integer. To make
use of this function, write first 'load("distrib")'.
-- Function: kurtosis_binomial (<n>,<p>)
Returns the kurtosis coefficient of a Binomial(n,p) random
variable, with 0 \leq p \leq 1 and n a positive integer. To make
use of this function, write first 'load("distrib")'.
-- Function: random_binomial (<n>,<p>)
random_binomial (<n>,<p>,<m>)
Returns a Binomial(n,p) random variate, with 0 \leq p \leq 1 and n
a positive integer. Calling 'random_binomial' with a third
argument <m>, a random sample of size <m> will be simulated.
The implemented algorithm is based on the one described in
Kachitvichyanukul, V. and Schmeiser, B.W. (1988) <Binomial Random
Variate Generation>. Communications of the ACM, 31, Feb., 216.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_poisson (<x>,<m>)
Returns the value at <x> of the probability function of a
Poisson(m) random variable, with m>0. To make use of this
function, write first 'load("distrib")'.
-- Function: cdf_poisson (<x>,<m>)
Returns the value at <x> of the distribution function of a
Poisson(m) random variable, with m>0.
(%i1) load ("distrib")$
(%i2) cdf_poisson(3,5);
(%o2) gamma_incomplete_regularized(4, 5)
(%i3) float(%);
(%o3) .2650259152973623
-- Function: quantile_poisson (<q>,<m>)
Returns the <q>-quantile of a Poisson(m) random variable, with m>0;
in other words, this is the inverse of 'cdf_poisson'. Argument <q>
must be an element of [0,1]. To make use of this function, write
first 'load("distrib")'.
-- Function: mean_poisson (<m>)
Returns the mean of a Poisson(m) random variable, with m>0. To
make use of this function, write first 'load("distrib")'.
-- Function: var_poisson (<m>)
Returns the variance of a Poisson(m) random variable, with m>0. To
make use of this function, write first 'load("distrib")'.
-- Function: std_poisson (<m>)
Returns the standard deviation of a Poisson(m) random variable,
with m>0. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_poisson (<m>)
Returns the skewness coefficient of a Poisson(m) random variable,
with m>0. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_poisson (<m>)
Returns the kurtosis coefficient of a Poisson random variable
Poi(m), with m>0. To make use of this function, write first
'load("distrib")'.
-- Function: random_poisson (<m>)
random_poisson (<m>,<n>)
Returns a Poisson(m) random variate, with m>0. Calling
'random_poisson' with a second argument <n>, a random sample of
size <n> will be simulated.
The implemented algorithm is the one described in Ahrens, J.H. and
Dieter, U. (1982) <Computer Generation of Poisson Deviates From
Modified Normal Distributions>. ACM Trans. Math. Software, 8, 2,
June,163-179.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_bernoulli (<x>,<p>)
Returns the value at <x> of the probability function of a
Bernoulli(p) random variable, with 0 \leq p \leq 1.
The Bernoulli(p) random variable is equivalent to the
Binomial(1,p).
(%i1) load ("distrib")$
(%i2) pdf_bernoulli(1,p);
(%o2) p
-- Function: cdf_bernoulli (<x>,<p>)
Returns the value at <x> of the distribution function of a
Bernoulli(p) random variable, with 0 \leq p \leq 1. To make use of
this function, write first 'load("distrib")'.
-- Function: quantile_bernoulli (<q>,<p>)
Returns the <q>-quantile of a Bernoulli(p) random variable, with 0
\leq p \leq 1; in other words, this is the inverse of
'cdf_bernoulli'. Argument <q> must be an element of [0,1]. To
make use of this function, write first 'load("distrib")'.
-- Function: mean_bernoulli (<p>)
Returns the mean of a Bernoulli(p) random variable, with 0 \leq p
\leq 1.
The Bernoulli(p) random variable is equivalent to the
Binomial(1,p).
(%i1) load ("distrib")$
(%i2) mean_bernoulli(p);
(%o2) p
-- Function: var_bernoulli (<p>)
Returns the variance of a Bernoulli(p) random variable, with 0 \leq
p \leq 1.
The Bernoulli(p) random variable is equivalent to the
Binomial(1,p).
(%i1) load ("distrib")$
(%i2) var_bernoulli(p);
(%o2) (1 - p) p
-- Function: std_bernoulli (<p>)
Returns the standard deviation of a Bernoulli(p) random variable,
with 0 \leq p \leq 1.
The Bernoulli(p) random variable is equivalent to the
Binomial(1,p).
(%i1) load ("distrib")$
(%i2) std_bernoulli(p);
(%o2) sqrt((1 - p) p)
-- Function: skewness_bernoulli (<p>)
Returns the skewness coefficient of a Bernoulli(p) random variable,
with 0 \leq p \leq 1.
The Bernoulli(p) random variable is equivalent to the
Binomial(1,p).
(%i1) load ("distrib")$
(%i2) skewness_bernoulli(p);
1 - 2 p
(%o2) ---------------
sqrt((1 - p) p)
-- Function: kurtosis_bernoulli (<p>)
Returns the kurtosis coefficient of a Bernoulli(p) random variable,
with 0 \leq p \leq 1.
The Bernoulli(p) random variable is equivalent to the
Binomial(1,p).
(%i1) load ("distrib")$
(%i2) kurtosis_bernoulli(p);
1 - 6 (1 - p) p
(%o2) ---------------
(1 - p) p
-- Function: random_bernoulli (<p>)
random_bernoulli (<p>,<n>)
Returns a Bernoulli(p) random variate, with 0 \leq p \leq 1.
Calling 'random_bernoulli' with a second 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_geometric (<x>,<p>)
Returns the value at <x> of the probability function of a
Geometric(p) random variable, with 0 < p <= 1.
The probability function is defined as p (1 - p)^x. This is
interpreted as the probability of x failures before the first
success.
'load("distrib")' loads this function.
-- Function: cdf_geometric (<x>,<p>)
Returns the value at <x> of the distribution function of a
Geometric(p) random variable, with 0 < p <= 1.
The probability from which the distribution function is derived is
defined as p (1 - p)^x. This is interpreted as the probability of
x failures before the first success.
'load("distrib")' loads this function.
-- Function: quantile_geometric (<q>,<p>)
Returns the <q>-quantile of a Geometric(p) random variable, with 0
< p <= 1; in other words, this is the inverse of 'cdf_geometric'.
Argument <q> must be an element of [0,1].
The probability from which the quantile is derived is defined as p
(1 - p)^x. This is interpreted as the probability of x failures
before the first success.
'load("distrib")' loads this function.
-- Function: mean_geometric (<p>)
Returns the mean of a Geometric(p) random variable, with 0 < p <=
1.
The probability from which the mean is derived is defined as p (1 -
p)^x. This is interpreted as the probability of x failures before
the first success.
'load("distrib")' loads this function.
-- Function: var_geometric (<p>)
Returns the variance of a Geometric(p) random variable, with 0 < p
<= 1.
The probability from which the variance is derived is defined as p
(1 - p)^x. This is interpreted as the probability of x failures
before the first success.
'load("distrib")' loads this function.
-- Function: std_geometric (<p>)
Returns the standard deviation of a Geometric(p) random variable,
with 0 < p <= 1.
The probability from which the standard deviation is derived is
defined as p (1 - p)^x. This is interpreted as the probability of
x failures before the first success.
'load("distrib")' loads this function.
-- Function: skewness_geometric (<p>)
Returns the skewness coefficient of a Geometric(p) random variable,
with 0 < p <= 1.
The probability from which the skewness is derived is defined as p
(1 - p)^x. This is interpreted as the probability of x failures
before the first success.
'load("distrib")' loads this function.
-- Function: kurtosis_geometric (<p>)
Returns the kurtosis coefficient of a geometric random variable
Geometric(p), with 0 < p <= 1.
The probability from which the kurtosis is derived is defined as p
(1 - p)^x. This is interpreted as the probability of x failures
before the first success.
'load("distrib")' loads this function.
-- Function: random_geometric (<p>)
random_geometric (<p>,<n>)
'random_geometric(<p>)' returns one random sample from a
Geometric(p) distribution, with 0 < p <= 1.
'random_geometric(<p>, <n>)' returns a list of <n> random samples.
The algorithm is based on simulation of Bernoulli trials.
The probability from which the random sample is derived is defined
as p (1 - p)^x. This is interpreted as the probability of x
failures before the first success.
'load("distrib")' loads this function.
-- Function: pdf_discrete_uniform (<x>,<n>)
Returns the value at <x> of the probability function of a Discrete
Uniform(n) random variable, with n a strictly positive integer. To
make use of this function, write first 'load("distrib")'.
-- Function: cdf_discrete_uniform (<x>,<n>)
Returns the value at <x> of the distribution function of a Discrete
Uniform(n) random variable, with n a strictly positive integer. To
make use of this function, write first 'load("distrib")'.
-- Function: quantile_discrete_uniform (<q>,<n>)
Returns the <q>-quantile of a Discrete Uniform(n) random variable,
with n a strictly positive integer; in other words, this is the
inverse of 'cdf_discrete_uniform'. Argument <q> must be an element
of [0,1]. To make use of this function, write first
'load("distrib")'.
-- Function: mean_discrete_uniform (<n>)
Returns the mean of a Discrete Uniform(n) random variable, with n a
strictly positive integer. To make use of this function, write
first 'load("distrib")'.
-- Function: var_discrete_uniform (<n>)
Returns the variance of a Discrete Uniform(n) random variable, with
n a strictly positive integer. To make use of this function, write
first 'load("distrib")'.
-- Function: std_discrete_uniform (<n>)
Returns the standard deviation of a Discrete Uniform(n) random
variable, with n a strictly positive integer. To make use of this
function, write first 'load("distrib")'.
-- Function: skewness_discrete_uniform (<n>)
Returns the skewness coefficient of a Discrete Uniform(n) random
variable, with n a strictly positive integer. To make use of this
function, write first 'load("distrib")'.
-- Function: kurtosis_discrete_uniform (<n>)
Returns the kurtosis coefficient of a Discrete Uniform(n) random
variable, with n a strictly positive integer. To make use of this
function, write first 'load("distrib")'.
-- Function: random_discrete_uniform (<n>)
random_discrete_uniform (<n>,<m>)
Returns a Discrete Uniform(n) random variate, with n a strictly
positive integer. Calling 'random_discrete_uniform' with a second
argument <m>, a random sample of size <m> 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_hypergeometric (<x>,<n1>,<n2>,<n>)
Returns the value at <x> of the probability function of a
Hypergeometric(n1,n2,n) random variable, with <n1>, <n2> and <n>
non negative integers and n<=n1+n2. Being <n1> the number of
objects of class A, <n2> the number of objects of class B, and <n>
the size of the sample without replacement, this function returns
the probability of event "exactly <x> objects are of class A".
To make use of this function, write first 'load("distrib")'.
-- Function: cdf_hypergeometric (<x>,<n1>,<n2>,<n>)
Returns the value at <x> of the distribution function of a
Hypergeometric(n1,n2,n) random variable, with <n1>, <n2> and <n>
non negative integers and n<=n1+n2. See 'pdf_hypergeometric' for a
more complete description.
To make use of this function, write first 'load("distrib")'.
-- Function: quantile_hypergeometric (<q>,<n1>,<n2>,<n>)
Returns the <q>-quantile of a Hypergeometric(n1,n2,n) random
variable, with <n1>, <n2> and <n> non negative integers and
n<=n1+n2; in other words, this is the inverse of
'cdf_hypergeometric'. Argument <q> must be an element of [0,1].
To make use of this function, write first 'load("distrib")'.
-- Function: mean_hypergeometric (<n1>,<n2>,<n>)
Returns the mean of a discrete uniform random variable
Hyp(n1,n2,n), with <n1>, <n2> and <n> non negative integers and
n<=n1+n2. To make use of this function, write first
'load("distrib")'.
-- Function: var_hypergeometric (<n1>,<n2>,<n>)
Returns the variance of a hypergeometric random variable
Hyp(n1,n2,n), with <n1>, <n2> and <n> non negative integers and
n<=n1+n2. To make use of this function, write first
'load("distrib")'.
-- Function: std_hypergeometric (<n1>,<n2>,<n>)
Returns the standard deviation of a Hypergeometric(n1,n2,n) random
variable, with <n1>, <n2> and <n> non negative integers and
n<=n1+n2. To make use of this function, write first
'load("distrib")'.
-- Function: skewness_hypergeometric (<n1>,<n2>,<n>)
Returns the skewness coefficient of a Hypergeometric(n1,n2,n)
random variable, with <n1>, <n2> and <n> non negative integers and
n<=n1+n2. To make use of this function, write first
'load("distrib")'.
-- Function: kurtosis_hypergeometric (<n1>,<n2>,<n>)
Returns the kurtosis coefficient of a Hypergeometric(n1,n2,n)
random variable, with <n1>, <n2> and <n> non negative integers and
n<=n1+n2. To make use of this function, write first
'load("distrib")'.
-- Function: random_hypergeometric (<n1>,<n2>,<n>)
random_hypergeometric (<n1>,<n2>,<n>,<m>)
Returns a Hypergeometric(n1,n2,n) random variate, with <n1>, <n2>
and <n> non negative integers and n<=n1+n2. Calling
'random_hypergeometric' with a fourth argument <m>, a random sample
of size <m> will be simulated.
Algorithm described in Kachitvichyanukul, V., Schmeiser, B.W.
(1985) <Computer generation of hypergeometric random variates.>
Journal of Statistical Computation and Simulation 22, 127-145.
To make use of this function, write first 'load("distrib")'.
-- Function: pdf_negative_binomial (<x>,<n>,<p>)
Returns the value at <x> of the probability function of a Negative
Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive
number. To make use of this function, write first
'load("distrib")'.
-- Function: cdf_negative_binomial (<x>,<n>,<p>)
Returns the value at <x> of the distribution function of a Negative
Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive
number.
(%i1) load ("distrib")$
(%i2) cdf_negative_binomial(3,4,1/8);
3271
(%o2) ------
524288
-- Function: quantile_negative_binomial (<q>,<n>,<p>)
Returns the <q>-quantile of a Negative Binomial(n,p) random
variable, with 0 < p \leq 1 and n a positive number; in other
words, this is the inverse of 'cdf_negative_binomial'. Argument
<q> must be an element of [0,1]. To make use of this function,
write first 'load("distrib")'.
-- Function: mean_negative_binomial (<n>,<p>)
Returns the mean of a Negative Binomial(n,p) random variable, with
0 < p \leq 1 and n a positive number. To make use of this
function, write first 'load("distrib")'.
-- Function: var_negative_binomial (<n>,<p>)
Returns the variance of a Negative Binomial(n,p) random variable,
with 0 < p \leq 1 and n a positive number. To make use of this
function, write first 'load("distrib")'.
-- Function: std_negative_binomial (<n>,<p>)
Returns the standard deviation of a Negative Binomial(n,p) random
variable, with 0 < p \leq 1 and n a positive number. To make use
of this function, write first 'load("distrib")'.
-- Function: skewness_negative_binomial (<n>,<p>)
Returns the skewness coefficient of a Negative Binomial(n,p) random
variable, with 0 < p \leq 1 and n a positive number. To make use
of this function, write first 'load("distrib")'.
-- Function: kurtosis_negative_binomial (<n>,<p>)
Returns the kurtosis coefficient of a Negative Binomial(n,p) random
variable, with 0 < p \leq 1 and n a positive number. To make use
of this function, write first 'load("distrib")'.
-- Function: random_negative_binomial (<n>,<p>)
random_negative_binomial (<n>,<p>,<m>)
Returns a Negative Binomial(n,p) random variate, with 0 < p \leq 1
and n a positive number. Calling 'random_negative_binomial' with a
third argument <m>, a random sample of size <m> will be simulated.
Algorithm described in Devroye, L. (1986) <Non-Uniform Random
Variate Generation>. Springer Verlag, p. 480.
To make use of this function, write first 'load("distrib")'.
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