public final class Cdf extends Object
Modifier and Type  Method and Description 

static double 
beta(double x,
double pin,
double qin)
Evaluates the beta cumulative probability distribution function.

static double 
betaMean(double pin,
double qin)
Evaluates the mean of the beta cumulative probability distribution function

static double 
betaVariance(double pin,
double qin)
Evaluates the variance of the beta cumulative probability distribution function

static double 
binomial(int k,
int n,
double pin)
Evaluates the binomial cumulative probability distribution function.

static double 
bivariateNormal(double x,
double y,
double rho)
Evaluates the bivariate normal cumulative probability distribution function.

static double 
chi(double chsq,
double df)
Evaluates the chisquared cumulative distribution function.

static double 
chiMean(double df)
Evaluates the mean of the chisquared cumulative probability distribution function

static double 
chiVariance(double df)
Evaluates the variance of the chisquared cumulative probability distribution function

static double 
complementaryChi(double chsq,
double df)
Calculates the complement of the chisquared cumulative distribution function.

static double 
complementaryF(double x,
double dfn,
double dfd)
Calculates the complement of the F distribution function.

static double 
complementaryF2(double x,
double dfn,
double dfd) 
static double 
complementaryNoncentralF(double f,
double df1,
double df2,
double lambda)
Calculates the complement of the noncentral F cumulative
distribution function.

static double 
complementaryStudentsT(double t,
double df)
Calculates the complement of the Student's t distribution.

static double 
discreteUniform(int x,
int n)
Evaluates the discrete uniform cumulative probability distribution function.

static double 
exponential(double x,
double scale)
Evaluates the exponential cumulative probability distribution function.

static double 
extremeValue(double x,
double mu,
double beta)
Evaluates the extreme value cumulative probability distribution function.

static double 
F(double x,
double dfn,
double dfd)
Evaluates the F cumulative probability distribution function.

static double 
gamma(double x,
double a)
Evaluates the gamma cumulative probability distribution function.

static double 
geometric(int x,
double pin)
Evaluates the discrete geometric cumulative probability distribution function.

static double 
hypergeometric(int k,
int sampleSize,
int defectivesInLot,
int lotSize)
Evaluates the hypergeometric cumulative probability distribution function.

static double 
logistic(double x,
double mu,
double s)
Evaluates the logistic cumulative probability distribution function.

static double 
logNormal(double x,
double mu,
double sigma)
Evaluates the standard lognormal cumulative probability distribution function.

static double 
noncentralBeta(double x,
double shape1,
double shape2,
double lambda)
Evaluates the noncentral beta cumulative distribution function (CDF).

static double 
noncentralchi(double chsq,
double df,
double alam)
Evaluates the noncentral chisquared cumulative probability distribution function.

static double 
noncentralF(double f,
double df1,
double df2,
double lambda)
Evaluates the noncentral F cumulative distribution function.

static double 
noncentralstudentsT(double t,
int idf,
double delta)
Evaluates the noncentral Student's t cumulative probability distribution function.

static double 
normal(double x)
Evaluates the normal (Gaussian) cumulative probability distribution function.

static double 
Pareto(double x,
double xm,
double k)
Evaluates the Pareto cumulative probability distribution function.

static double 
poisson(int k,
double theta)
Evaluates the Poisson cumulative probability distribution function.

static double 
Rayleigh(double x,
double alpha)
Evaluates the Rayleigh cumulative probability distribution function.

static double 
studentsT(double t,
double df)
Evaluates the Student's t cumulative probability distribution function.

static double 
uniform(double x,
double aa,
double bb)
Evaluates the uniform cumulative probability distribution function.

static double 
Weibull(double x,
double gamma,
double alpha)
Evaluates the Weibull cumulative probability distribution function.

public static double beta(double x, double pin, double qin)
Method beta
evaluates the distribution function of a
beta random variable with parameters pin
and qin
.
This function is sometimes called the incomplete beta ratio and,
with p = pin
and q = qin
, is
denoted by . It is given by
where is the gamma function. The value of the distribution function is the probability that the random variable takes a value less than or equal to x.
The integral in the expression above is called the incomplete beta function and is denoted by . The constant in the expression is the reciprocal of the beta function (the incomplete beta function evaluated at x=1) and is denoted by .
beta
uses the method of Bosten and Battiste (1974).
x
 a double
, the argument at which the function is to be evaluated.pin
 a double
, the first beta distribution parameter.qin
 a double
, the second beta distribution parameter.double
, the probability that a beta random variable takes
on a value less than or equal to x
.public static double betaMean(double pin, double qin)
pin
 a double
, the first beta distribution parameter.qin
 a double
, the second beta distribution parameter.double
, the mean of the beta distribution
function.public static double betaVariance(double pin, double qin)
pin
 a double
, the first beta distribution parameter.qin
 a double
, the second beta distribution parameter.double
, the variance of the beta distribution
function.public static double binomial(int k, int n, double pin)
Method binomial
evaluates the distribution function of
a binomial random variable with parameters n and
p with p=pin. It does this by summing probabilities of the
random variable taking on the specific values in its range. These
probabilities are computed by the recursive relationship
To avoid the possibility of underflow, the probabilities are computed
forward from 0, if k is not
greater than n times p, and
are computed backward from n, otherwise. The
smallest positive machine number, ,
is used as the starting value for summing the probabilities, which are
rescaled by if forward
computation is performed and by if
backward computation is done. For the special case of
p = 0, binomial
is set to
1; and for the case p = 1,
binomial
is set to 1 if
k = n and to 0 otherwise.
k
 the int
argument for which the binomial distribution function
is to be evaluated.n
 the int
number of Bernoulli trials.pin
 a double
scalar value representing the
probability of success on each independent trial.double
scalar value representing the probability
that a binomial random variable takes a value less than or
equal to k
. This value is the probability that k
or fewer successes
occur in n
independent Bernoulli trials, each of which
has a pin
probability of success.public static double bivariateNormal(double x, double y, double rho)
x
 is the xcoordinate of the point for which the
bivariate normal distribution function is to be evaluated.y
 is the ycoordinate of the point for which the
bivariate normal distribution function is to be evaluated.rho
 is the correlation coefficient.rho
satisfies
and
.public static double chi(double chsq, double df)
Method chi
evaluates the cumulative distribution function
(CDF) F, of a chisquared random variable with
df
degrees of freedom, that is, with x = chsq
and = df
,
where is the gamma function. The value is the probability that the random variable takes a value less than or equal to x.
For , chi
uses the
WilsonHilferty approximation (Abramowitz and Stegun [A&S] 1964, equation 26.4.17)
for p in terms of the normal CDF, which is evaluated using method
normal
.
For , chi
uses series
expansions to evaluate p: for ,
chi
calculates p using A&S equation 6.5.29, and for
, chi
calculates p
using the continued fraction (CF) expansion of the incomplete gamma function given in
A&S equation 6.5.31 and implemented using Lentz's algorithm (Lentz, W.J., 1976,
Applied Optics, vol. 15, pp. 668671) as modified by Thompson & Barnett
(Thompson, I.J., and Barnett, A.R., 1986, Journal of Computational Physics,
vol. 64, pp. 490509).
For greater right tail accuracy, see complementaryChi(double, double)
.
chsq
 a double
scalar value representing the argument
at which the function is to be evaluated.df
 a double
scalar value representing the number
of degrees of freedom. df
must be positive.double
scalar value representing the
probability that a chisquared random variable
takes a value less than or equal to chsq
.public static double chiMean(double df)
df
 a double
scalar value representing the number
of degrees of freedom. This must be at least 0.5.double
, the mean of the chisquared distribution
function.public static double chiVariance(double df)
df
 a double
scalar value representing the number
of degrees of freedom. This must be at least 0.5.double
, the variance of the chisquared distribution
function.public static double complementaryChi(double chsq, double df)
Method complementaryChi
evaluates the cumulative distribution function,
, of a chisquared random variable with
df
degrees of freedom, that is, with x = chsq
and = df
,
where is the gamma function.
The value of the complementaryChi
distribution function at
the point x, , is the probability that
the random variable takes a value greater than x.
For , complementaryChi
uses the WilsonHilferty
approximation (Abramowitz and Stegun [A&S] 1964, equation 26.4.17) for p
in terms of the normal CDF, which is evaluated using method normal
.
For , complementaryChi
uses series
expansions to evaluate p: for ,
complementaryChi
calculates p using A&S series 6.5.29, and for
, complementaryChi
calculates p
using the continued fraction expansion of the incomplete gamma function given in
A&S equation 6.5.31 and implemented using Lentz's algorithm (Lentz, W.J., 1976,
Applied Optics, vol. 15, pp. 668671) as modified by Thompson & Barnett
(Thompson, I.J., and Barnett, A.R., 1986, Journal of Computational Physics,
vol. 64, pp. 490509).
complementaryChi
provides greater right tail accuracy for the
Chisquared distribution than does 1  chi
.
chsq
 a double
scalar value at which the function
is to be evaluated.df
 a double
scalar value representing the
number of degrees of freedom. df
must be positive.double
scalar value representing the
probability that a chisquared random variable
takes a value greater than chsq
.public static double complementaryF(double x, double dfn, double dfd)
complementaryF
evaluates one minus the distribution
function of a Snedecor's F random variable with dfn
numerator degrees of freedom and dfd
denominator degrees of
freedom. The function is evaluated by making a transformation to a beta
random variable and then using the function beta
. If
X is an F variate with and
degrees of freedom and
, then Y
is a beta variate with parameters and
. complementaryF
also uses a
relationship between F random variables that can be expressed as
follows:
This function provides higher right tail accuracy for the F distribution.
x
 a double
, the argument at which is to be
evaluated.dfn
 a double
, the numerator degrees of freedom.
It must be positive.dfd
 a double
, the denominator degrees of
freedom. It must be positive.double
, the probability that an F
random variable takes on a value greater than
x
.public static double complementaryF2(double x, double dfn, double dfd)
public static double complementaryNoncentralF(double f, double df1, double df2, double lambda)
The complementary noncentral F distribution is a generalization of the complementary F distribution. If is a noncentral chisquare random variable with noncentrality parameter and degrees of freedom, and is a chisquare random variable with degrees of freedom which is statistically independent of , then
is a noncentral Fdistributed random variable whose CDF is given by: where: and is the gamma function, =df1
,
= df2
, = lambda
,
and f = f
. The above series expansion for the
noncentral F was taken from Butler and Paolella
(1999) (see
Paolella.pdf), with the correction for the recursion relation
given below:
extracted from the AS 63 algorithm for calculating the incomplete beta
function as described by Majumder and Bhattacharjee (1973).
The series approximation of the complementary (cmp) noncentral F
CDF, denoted by , is obtainable by
using the following identities:
Thus:
We can use the above expansion of and the identities:
to recursively calculate .
With a noncentrality parameter of zero, the noncentral F distribution is the same as the F distribution.
f
 a double
value representing the argument at which
the function is to be evaluated. f
must be nonnegative.df1
 a double
value representing the number of
numerator degrees of freedom. df1
must be positive.df2
 a double
value representing the number of
denominator degrees of freedom. df2
must be positive.lambda
 a double
value representing the
noncentrality parameter. lambda
must be nonnegative.double
scalar value representing the probability
that a noncentral F random variable takes a value greater than
f
.public static double complementaryStudentsT(double t, double df)
Method complementaryStudentsT
evaluates one minus the
distribution function of a Student's t random variable with
df
degrees of freedom. If the square of t is greater
than or equal to df
, the relationship of a
t to an f random variable (and
subsequently, to a beta random variable) is exploited, and routine
beta
is used. Otherwise, the method described by Hill
(1970) is used. If df
is not an integer, if df
is greater than 19, or if df
is greater than 200, a
CornishFisher expansion is used to evaluate the distribution function.
If df
is less than 20 and
is less than
2.0, a trigonometric series (see Abramowitz and
Stegun 1964, equations 26.7.3 and 26.7.4, with some rearrangement) is
used. For the remaining cases, a series given by Hill (1970) that
converges well for large values of t
is used.
This function provides higher right tail accuracy for the Student's t distribution.
t
 a double
scalar value for which is to be evaluateddf
 a double
scalar value representing the number
of degrees of freedom. This must be at least one.double
scalar value representing the
probability that a Student's t random variable
takes a value greater than t
.public static double discreteUniform(int x, int n)
x
 an int
scalar value representing the argument
at which the function is to be evaluated.
x
should be a value between the lower
limit 0 and upper limit n
n
 an int
scalar value representing the
upper limit of the discrete uniform distribution.double
scalar value representing the probability
that a discrete uniform random variable takes a value
less than or equal to x
.public static double exponential(double x, double scale)
Method exponential
is a special case of the gamma
distribution function, which evaluates the distribution function,
F, with scale parameter b and shape parameter a,
used in the gamma distribution function equal to 1.0. That is,
where is the gamma function. (The gamma function is the integral from 0 to of the same integrand as above). The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.
If x
is less than or equal to 1.0,
gamma
uses a series expansion. Otherwise, a continued
fraction expansion is used. (See Abramowitz and Stegun, 1964.)
x
 a double
scalar value representing the
argument at which the function is to be evaluated.scale
 a double
scalar value representing the scale
parameter, .double
scalar value representing the
probability that an exponential random variable takes
on a value less than or equal to x
.public static double extremeValue(double x, double mu, double beta)
Method extremeValue
, also known as the Gumbel minimum
distribution, evaluates the extreme value distribution function,
F, of a uniform random variable with location parameter
and shape parameter ; that is,
The case where and is called the standard Gumbel distribution.
Random numbers are generated by evaluating uniform variates , equating the continuous distribution function, and then solving for by first computing .
x
 a double
scalar value representing the
argument at which the function is to be evaluated.mu
 a double
scalar value representing the
location parameter, .beta
 a double
scalar value representing the scale
parameter, double
scalar value representing the
probability that an extreme value random variable takes
on a value less than or equal to x
.public static double F(double x, double dfn, double dfd)
F
evaluates the distribution function of a Snedecor's F
random variable with dfn
numerator degrees of freedom and
dfd
denominator degrees of freedom. The function is
evaluated by making a transformation to a beta random variable and then
using the function beta
. If X is an
F variate with and
degrees of freedom and
, then Y
is a beta variate with parameters and
. F
also uses a relationship
between F random variables that can be expressed as
follows:
For greater right tail accuracy, see complementaryF(double, double, double)
.
x
 a double
, the argument at which
the function is to be evaluated.dfn
 a double
, the numerator degrees
of freedom. It must be positive.dfd
 a double
, the denominator degrees
of freedom. It must be positive.double
, the probability that an
F random variable takes on a value less than or
equal to x
.public static double gamma(double x, double a)
Method gamma
evaluates the distribution function,
F, of a gamma random variable with shape parameter
a; that is,
where is the gamma function. (The gamma function is the integral from 0 to of the same integrand as above). The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.
The gamma distribution is often defined as a twoparameter distribution with a scale parameter b (which must be positive), or even as a threeparameter distribution in which the third parameter c is a location parameter. In the most general case, the probability density function over is
If T is such a random variable with parameters
a, b, and c,
the probability that can be obtained from
gamma
by setting .
If X is less than a or if
X is less than or equal to 1.0,
gamma
uses a series expansion. Otherwise, a continued
fraction expansion is used. (See Abramowitz and Stegun, 1964.)
x
 a double
scalar value representing the
argument at which the function is to be evaluated.a
 a double
scalar value representing the
shape parameter. This must be positive.double
scalar value representing the
probability that a gamma random variable takes
on a value less than or equal to x
.public static double geometric(int x, double pin)
x
 an int
scalar value representing the argument
at which the function is to be evaluatedpin
 an double
scalar value representing the probability parameter
for each independent trial (the probability of success for each
independent trial).double
scalar value representing the probability
that a geometric random variable takes a value
less than or equal to x
. The return value is the
probability that up to x
trials would be observed
before observing a success.public static double hypergeometric(int k, int sampleSize, int defectivesInLot, int lotSize)
Method hypergeometric
evaluates the distribution
function of a hypergeometric random variable with parameters
n, l, and m.
The hypergeometric random variable X can be thought
of as the number of items of a given type in a random sample of size
n that is drawn without replacement from a
population of size l containing m
items of this type. The probability function is
where .
If k is greater than or equal to
i and less than or equal to ,
hypergeometric
sums the terms in this expression for
j going from i up to
k. Otherwise, hypergeometric
returns
0 or 1, as appropriate. So, as
to avoid rounding in the accumulation, hypergeometric
performs the summation differently depending on whether or not
k is greater than the mode of the distribution,
which is the greatest integer less than or equal to
.
k
 an int
, the argument at which
the function is to be evaluated.sampleSize
 an int
, the sample size, n
.defectivesInLot
 an int
, the number of defectives
in the lot, m
.lotSize
 an int
, the lot size, l
.double
, the probability that a
hypergeometric random variable takes a value
less than or equal to k
.public static double logistic(double x, double mu, double s)
Method logistic
evaluates the distribution function, F, of
a logistic random variable with location parameter and
scale parameter s. It is given by
where .
x
 a double
scalar value representing the argument
at which the function is to be evaluated.mu
 a double
scalar value representing the
location parameter, .s
 a double
scalar value representing the
scale parameter.double
scalar value representing the probability
that a logistic random variable takes a value
less than or equal to x
.public static double logNormal(double x, double mu, double sigma)
x
 a double
scalar value representing the argument
at which the function is to be evaluated.mu
 a double
scalar value representing the location parameter.sigma
 a double
scalar value representing the shape parameter.
sigma
must be a positive.double
scalar value representing the probability
that a standard lognormal random variable
takes a value less than or equal to x
.public static double noncentralBeta(double x, double shape1, double shape2, double lambda)
The noncentral beta distribution is a generalization of the beta distribution. If is a noncentral chisquare random variable with noncentrality parameter and degrees of freedom, and is a chisquare random variable with degrees of freedom which is statistically independent of , then
is a noncentral betadistributed random variable and
is a noncentral Fdistributed random variable. The CDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F CDF:
where is the noncentral beta CDF with = x
,
= shape1
, = shape2
, and noncentrality parameter
= lambda
; is the noncentral F CDF
with argument f, numerator and denominator degrees of freedom and
respectively, and noncentrality parameter ; and:
(See documentation for class Cdf
method noncentralF
for a discussion of how the noncentral F CDF
is defined and calculated.)
With a noncentrality parameter of zero, the noncentral beta distribution is the same as the beta distribution.
x
 a double
scalar value representing the argument
at which the function is to be evaluated.
x
must be nonnegative and less than or equal to 1.shape1
 a double
scalar value representing the first
shape parameter. shape1
must be positive.shape2
 a double
scalar value representing the second
shape parameter. shape2
must be positive.lambda
 a double
scalar value representing the noncentrality parameter.
lambda
must nonnegative.double
scalar value representing the probability that a
noncentral beta random variable takes a value less than or equal to x
.public static double noncentralchi(double chsq, double df, double alam)
Method noncentralchi
evaluates the distribution function,
F, of a noncentral chisquared random variable with
df
degrees of freedom and noncentrality parameter alam
, that is, with
, , and ,
where is the gamma function. This is a series of central chisquared distribution functions with Poisson weights. The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.
The noncentral chisquared random variable can be defined by the distribution function above, or alternatively and equivalently, as the sum of squares of independent normal random variables. If the have independent normal distributions with means and variances equal to one and
then has a noncentral chisquared distribution with degrees of freedom and noncentrality parameter equal to
With a noncentrality parameter of zero, the noncentral chisquared distribution is the same as the chisquared distribution.
noncentralchi
determines the point at which the Poisson weight is greatest, and then
sums forward and backward from that point, terminating when the additional terms are sufficiently
small or when a maximum of 1000 terms have been accumulated. The recurrence relation 26.4.8 of
Abramowitz and Stegun (1964) is used to speed the evaluation of the central chisquared distribution
functions.
chsq
 a double
scalar value representing the argument
at which the function is to be evaluated.df
 a double
scalar value representing the number
of degrees of freedom. df
must be positive.alam
 a double
scalar value representing the noncentrality
parameter. This must be nonnegative, and alam + df
must be less than or equal to 200,000.double
scalar value representing the
probability that a chisquared random variable
takes a value less than or equal to chsq
.public static double noncentralF(double f, double df1, double df2, double lambda)
The noncentral F distribution is a generalization of the F distribution. If is a noncentral chisquare random variable with noncentrality parameter and degrees of freedom, and is a chisquare random variable with degrees of freedom which is statistically independent of , then
is a noncentral Fdistributed random variable whose CDF is given by: where: and is the gamma function, =df1
,
= df2
, = lambda
,
and f = f
.
With a noncentrality parameter of zero, the noncentral F distribution is the same as the F distribution.
f
 a double
value representing the argument
at which the function is to be evaluated. f
must be nonnegative.df1
 a double
value representing the number
of numerator degrees of freedom. df1
must be positive.df2
 a double
value representing the number
of denominator degrees of freedom. df2
must
be positive.lambda
 a double
value representing the noncentrality
parameter. lambda
must be nonnegative.double
scalar value representing the
probability that a noncentral F random variable
takes a value less than or equal to f
.public static double noncentralstudentsT(double t, int idf, double delta)
Method noncentralstudentsT
evaluates the distribution function
F
of a noncentral t random variable with idf
degrees of freedom and noncentrality parameter delta
; that is, with
, , and ,
where is the gamma function. The value of the distribution function at the point is the probability that the random variable takes a value less than or equal to .
The noncentral t random variable can be defined by the distribution function above, or alternatively and equivalently, as the ratio of a normal random variable and an independent chisquared random variable. If w has a normal distribution with mean and variance equal to one, has an independent chisquared distribution with degrees of freedom, and
then has a noncentral distribution with degrees of freedom and noncentrality parameter .
The distribution function of the noncentral can also be expressed as a
double integral involving a normal density function (see, for example, Owen 1962, page 108). The method
noncentralstudentsT
uses the method of Owen (1962, 1965), which uses repeated integration by parts
on that alternate expression for the distribution function.
t
 a double
scalar value representing the argument
at which the function is to be evaluated.idf
 an int
scalar value representing the number of
degrees of freedom. This must be positive.delta
 a double
scalar value representing the noncentrality
parameter.double
scalar value representing the probability
that a noncentral Student's t random variable takes a value less than or
equal to t
.public static double normal(double x)
Method normal
evaluates the distribution function,
, of a standard normal (Gaussian) random
variable, that is,
The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.
The standard normal distribution (for which normal
is
the distribution function) has mean of 0 and
variance of 1. The probability that a normal
random variable with mean and variance
is less than y
is given by normal
evaluated at
.
is evaluated by use of the complementary error function, erfc. The relationship is:
x
 a double
scalar value representing the argument
at which the function is to be evaluated.double
scalar value representing the probability
that a normal variable takes a value less than or equal to x
.public static double Pareto(double x, double xm, double k)
Method Pareto
evaluates the distribution function, F, of
a Pareto random variable with scale parameter and
shape parameter k. It is given by
where and . The function is only defined for
x
 a double
scalar value representing the argument
at which the function is to be evaluated.xm
 a double
scalar value representing the
scale parameter, .k
 a double
scalar value representing the
shape parameter, .double
scalar value representing the probability
that a Pareto random variable takes a value
less than or equal to x
.public static double poisson(int k, double theta)
poisson
evaluates the distribution function of a Poisson
random variable with parameter theta
. theta
,
which is the mean of the Poisson random variable, must be positive. The
probability function (with ) is
The individual terms are calculated from the tails of the
distribution to the mode of the distribution and summed.
poisson
uses the recursive relationship
with .
k
 the int
argument for which the Poisson distribution function
is to be evaluated.theta
 a double
scalar value representing the mean of the Poisson distribution.double
scalar value representing the probability that a Poisson random
variable takes a value less than or equal to k.public static double Rayleigh(double x, double alpha)
Method Rayleigh
is a special case of Weibull distribution
function where the shape parameter gamma
is 2.0; that is,
where alpha
is the scale parameter.
x
 a double
scalar value representing the argument
at which the function is to be evaluated. It must be nonnegative.alpha
 a double
scalar value representing the scale parameter.double
scalar value representing the probability
that a Rayleigh random variable takes a value less than or
equal to x
.public static double studentsT(double t, double df)
Method studentsT
evaluates the distribution function of
a Student's t random variable with df
degrees of freedom. If the square of t is greater
than or equal to df
, the relationship of a
t to an f random variable (and
subsequently, to a beta random variable) is exploited, and routine
beta
is used. Otherwise, the method described by Hill
(1970) is used. If df
is not an integer, if df
is greater than 19, or if df
is greater than 200, a
CornishFisher expansion is used to evaluate the distribution function.
If df
is less than 20 and
is less than
2.0, a trigonometric series (see Abramowitz and
Stegun 1964, equations 26.7.3 and 26.7.4, with some rearrangement) is
used. For the remaining cases, a series given by Hill (1970) that
converges well for large values of t
is used.
For greater right tail accuracy, see complementaryStudentsT(double, double)
.
t
 a double
scalar value representing the argument
at which the function is to be evaluateddf
 a double
scalar value representing the number of
degrees of freedom. This must be at least one.double
scalar value representing the probability
that a Student's t random variable takes a value less than or
equal to t
.public static double uniform(double x, double aa, double bb)
Method uniform
evaluates the distribution function,
F, of a uniform random variable with location parameter
aa and scale parameter bb; that is,
x
 a double
scalar value representing the argument
at which the function is to be evaluated.aa
 a double
scalar value representing the location parameter.bb
 a double
scalar value representing the scale parameter.double
scalar value representing the probability
that a uniform random variable takes a value less than or
equal to x
.public static double Weibull(double x, double gamma, double alpha)
Method Weibull
evaluates the distribution function given by
x
 a double
scalar value representing the argument
at which the function is to be evaluated. It must be nonnegative.gamma
 a double
scalar value representing the shape parameter, .alpha
 a double
scalar value representing the scale parameter, .double
scalar value representing the probability
that a Weibull random variable takes a value less than or
equal to x
.Copyright © 19702015 Rogue Wave Software
Built October 13 2015.