help density functions
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Title
[D] functions -- Functions
Description
This is a quick reference for the probability distributions and density
functions. For help on all functions, see [D] functions.
Probability distributions and density functions
The probability distribution and density functions are organized under
the following headings:
Beta and noncentral beta distributions
Binomial distribution
Chi-squared and noncentral chi-squared distributions
F and noncentral F distributions
Gamma distribution
Hypergeometric distribution
Negative binomial distribution
Normal (Gaussian), log of the normal, and binormal distributions
Poisson distribution
Student's t distribution
Random-number functions for random-number generators
Beta and noncentral beta distributions
ibeta(a,b,x)
Domain a: 1e-10 to 1e+17
Domain b: 1e-10 to 1e+17
Domain x: -8e+307 to 8e+307
Interesting domain is 0 < x < 1
Range: 0 to 1
Description: returns the cumulative beta distribution with shape
parameters a and b
returns 0 if x < 0.
returns 1 if x > 1.
ibeta() returns the regularized incomplete beta
function, also known as the incomplete beta function
ratio. The incomplete beta function without
regularization is given by
(gamma(a)*gamma(b)/gamma(a+b))*ibeta(a,b,x)
or, better when a or b might be large,
exp(lngamma(a)+lngamma(b)-lngamma(a+b))*ibeta(a,b,x).
Here is an example of the use of the regularized
incomplete beta function. Although Stata has a
cumulative binomial function (see binomial()), the
probability that an event occurs k or fewer times in n
trials, when the probability of one event is p, can be
evaluated as cond(k==n,1,1-ibeta(k+1,n-k,p)). The
reverse cumulative binomial (the probability that an
event occurs k or more times) can be evaluated as
cond(k==0,1,ibeta(k,n-k+1,p)).
betaden(a,b,x)
Domain a: 1e-323 to 8e+307
Domain b: 1e-323 to 8e+307
Domain x: 1e-323 to 8e+307
Interesting domain is 0 < x < 1
Range: 0 to 8e+307
Description: returns the probability density of the beta
distribution, where a and b are shape parameters.
returns 0 if x < 0 or x > 1.
ibetatail(a,b,x)
Domain a: 1e-10 to 1e+17
Domain b: 1e-10 to 1e+17
Domain x: -8e+307 to 8e+307
Interesting domain is 0 < x < 1
Range: 0 to 1
Description: returns the reverse cumulative (upper-tail, survival)
beta distribution with shape parameters a and b.
returns 1 if x < 0.
returns 0 if x > 1.
ibetatail() is also known as the complement to the
incomplete beta function (ratio).
invibeta(a,b,p)
Domain a: 1e-10 to 1e+17
Domain b: 1e-10 to 1e+17
Domain p: 0 to 1
Range: 0 to 1
Description: returns the inverse cumulative beta distribution: if
ibeta(a,b,x) = p, then invibeta(a,b,p) = x.
invibetatail(a,b,p)
Domain a: 1e-10 to 1e+17
Domain b: 1e-10 to 1e+17
Domain p: 0 to 1
Range: 0 to 1
Description: returns the inverse reverse cumulative (upper-tail,
survival) beta distribution: if ibetatail(a,b,x) =
p, then invibetatail(a,b,p) = x.
nibeta(a,b,L,x)
Domain a: 1e-323 to 8e+307
Domain b: 1e-323 to 8e+307
Domain L: 0 to 1,000
Domain x: -8e+307 to 8e+307
Interesting domain is 0 < x < 1
Range: 0 to 1
Description: returns the cumulative noncentral beta distribution,
where a and b are shape parameters, L is the
noncentrality parameter, and x is the value of a
beta random variable.
returns 0 if x < 0.
returns 1 if x > 1.
nibeta(a,b,0,x) = ibeta(a,b,x), but ibeta() is the
preferred function to use for the central beta
distribution. nibeta() is computed using an algorithm
described in Johnson, Kotz, and Balakrishnan (1995).
nbetaden(a,b,L,x)
Domain a: 1e-323 to 8e+307
Domain b: 1e-323 to 8e+307
Domain L: 0 to 1,000
Domain x: -8e+307 to 8e+307
Interesting domain is 0 < x < 1
Range: 0 to 8e+307
Description: returns the probability density function of the
noncentral beta distribution, where a and b are
shape parameters, L is the noncentrality parameter,
and x is the value of a beta random variable.
returns 0 if x < 0 or x > 1.
nbetaden(a,b,0,x)= betaden(a,b,x), but betaden() is the
preferred function to use for the central beta
distribution. nbetaden() is computed using an algorithm
described in Johnson, Kotz, and Balakrishnan (1995).
invnibeta(a,b,L,p)
Domain a: 1e-323 to 8e+307
Domain b: 1e-323 to 8e+307
Domain L: 0 to 1,000
Domain p: 0 to 1
Range: 0 to 1
Description: returns the inverse cumulative noncentral beta
distribution: if nibeta(a,b,L,x) = p, then
invnibeta(a,b,L,p) = x.
Binomial distribution
binomial(n,k,p)
Domain n: 0 to 1e+17
Domain k: -8e+307 to 8e+307
Interesting domain is 0 < k < n
Domain p: 0 to 1
Range: 0 to 1
Description: returns the probability of observing floor(k) or fewer
successes in floor(n) trials when the probability of
a success on one trial is p.
returns 0 if k < 0.
returns 1 if k > n.
binomialp(n,k,p)
Domain n: 1 to 1e+6
Domain k: 0 to n
Domain p: 0 to 1
Range: 0 to 1
Description: returns the probability of observing floor(k) successes
in floor(n) trials when the probability of a success
on one trial is p.
binomialtail(n,k,p)
Domain n: 0 to 1e+17
Domain k: -8e+307 to 8e+307
Interesting domain is 0 < k < n
Domain p: 0 to 1
Range: 0 to 1
Description: returns the probability of observing floor(k) or more
successes in floor(n) trials when the probability of
a success on one trial is p.
returns 1 if k < 0.
returns 0 if k > n.
invbinomial(n,k,p)
Domain n: 1 to 1e+17
Domain k: 0 to n - 1
Domain p: 0 to 1 (exclusive)
Range: 0 to 1
Description: returns the inverse of the cumulative binomial; i.e., it
returns the probability of success on one trial such
that the probability of observing floor(k) or fewer
successes in floor(n) trials is p.
invbinomialtail(n,k,p)
Domain n: 1 to 1e+17
Domain k: 1 to n
Domain p: 0 to 1 (exclusive)
Range: 0 to 1
Description: returns the inverse of the right cumulative binomial;
i.e., it returns the probability of success on one
trial such that the probability of observing
floor(k) or more successes in floor(n) trials is p.
Chi-squared and noncentral chi-squared distributions
chi2(n,x)
Domain n: 2e-10 to 2e+17 (may be nonintegral)
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: 0 to 1
Description: returns the cumulative chi-squared distribution with n
degrees of freedom, chi2(n,x) = gammap(n/2,x/2).
returns 0 if x < 0.
chi2tail(n,x)
Domain n: 2e-10 to 2e+17 (may be nonintegral)
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: 0 to 1
Description: returns the reverse cumulative (upper-tail, survival)
chi-squared distribution with n degrees of freedom.
chi2tail(n,x) = 1 - chi2(n,x)
returns 1 if x < 0.
invchi2(n,p)
Domain n: 2e-10 to 2e+17 (may be nonintegral)
Domain p: 0 to 1
Range: 0 to 8e+307
Description: returns the inverse of chi2(): if chi2(n,x) = p, then
invchi2(n,p) = x.
invchi2tail(n,p)
Domain n: 2e-10 to 2e+17 (may be nonintegral)
Domain p: 0 to 1
Range: 0 to 8e+307
Description: returns the inverse of chi2tail(): if chi2tail(n,x) = p,
then invchi2tail(n,p) = x.
nchi2(n,L,x)
Domain n: integers 1 to 200
Domain L: 0 to 1,000
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: 0 to 1
Description: returns the cumulative noncentral chi-squared
distribution, where n denotes the degrees of
freedom, L is the noncentrality parameter, and x is
the value of chi-squared.
returns 0 if x < 0.
nchi2(n,0,x) = chi2(n,x), but chi2() is the preferred
function to use for the central chi-squared
distribution. nchi2() is computed using the algorithm
of Haynam, Govindarajulu, and Leone (1970).
invnchi2(n,L,p)
Domain n: integers 1 to 200
Domain L: 0 to 1,000
Domain p: 0 to 1
Range: 0 to 8e+307
Description: returns the inverse cumulative noncentral chi-squared
distribution: if nchi2(n,L,x) = p, then
invnchi2(n,L,p) = x; n must be an integer.
npnchi2(n,x,p)
Domain n: integers 1 to 200
Domain x: 0 to 8e+307
Domain p: 1e-138 to 1 - 2^(-52)
Range: 0 to 1,000
Description: returns the noncentrality parameter, L, for the
noncentral chi-squared: if nchi2(n,L,x) = p, then
npnchi2(n,x,p) = L.
F and noncentral F distributions
F(n1,n2,f)
Domain n1: 2e-10 to 2e+17 (may be nonintegral)
Domain n2: 2e-10 to 2e+17 (may be nonintegral)
Domain f: -8e+307 to 8e+307
Interesting domain is f > 0
Range: 0 to 1
Description: returns the cumulative F distribution with n1 numerator
and n2 denominator degrees of freedom.
returns 0 if f < 0.
Fden(n1,n2,f)
Domain n1: 1e-323 to 8e+307 (may be nonintegral)
Domain n2: 1e-323 to 8e+307 (may be nonintegral)
Domain f: -8e+307 to 8e+307
Interesting domain is f > 0
Range: 0 to 8e+307
Description: returns the probability density function for the F
distribution with n1 numerator and n2 denominator
degrees of freedom.
returns 0 if f < 0.
Ftail(n1,n2,f)
Domain n1: 2e-10 to 2e+17 (may be nonintegral)
Domain n2: 2e-10 to 2e+17 (may be nonintegral)
Domain f: -8e+307 to 8e+307
Interesting domain is f > 0
Range: 0 to 1
Description: returns the reverse cumulative (upper-tail, survival) F
distribution with n1 numerator and n2 denominator
degrees of freedom. Ftail(n1,n2,f) = 1 - F(n1,n2,f)
returns 1 if f < 0.
invF(n1,n2,p)
Domain n1: 2e-10 to 2e+17 (may be nonintegral)
Domain n2: 2e-10 to 2e+17 (may be nonintegral)
Domain p: 0 to 1
Range: 0 to 8e+307
Description: returns the inverse cumulative F distribution: if
F(n1,n2,f) = p, then invF(n1,n2,p) = f.
invFtail(n1,n2,p)
Domain n1: 2e-10 to 2e+17 (may be nonintegral)
Domain n2: 2e-10 to 2e+17 (may be nonintegral)
Domain p: 0 to 1
Range: 0 to 8e+307
Description: returns the inverse reverse cumulative (upper-tail,
survival) F distribution: if Ftail(n1,n2,f) = p,
then invFtail(n1,n2,p) = f.
nFden(n1,n2,L,x)
Domain n1: 1e-323 to 8e+307 (may be nonintegral)
Domain n2: 1e-323 to 8e+307 (may be nonintegral)
Domain L: 0 to 1,000
Domain x: -8e+307 to 8e+307
Interesting domain is f > 0
Range: 0 to 8e+307
Description: returns the probability density function of the
noncentral F density with n1 numerator and n2
denominator degrees of freedom and noncentrality
parameter L.
returns 0 if f < 0.
nFden(n1,n2,0,F) = Fden(n1,n2,F), but Fden() is the
preferred function to use for the central F
distribution.
Also, if F follows the noncentral F distribution with n1
and n2 degrees of freedom and noncentrality parameter L,
then
n1 F
---------
n2 + n1 F
follows a noncentral beta distribution with shape
parameters a=v1/2, b=v2/2, and noncentrality parameter
L, as given in nbetaden(). nFden() is computed based on
this relationship.
nFtail(n1,n2,L,f)
Domain n1: 1e-323 to 8e+307 (may be nonintegral)
Domain n2: 1e-323 to 8e+307 (may be nonintegral)
Domain L: 0 to 1,000
Domain f: -8e+307 to 8e+307
Interesting domain is f > 0
Range: 0 to 1
Description: returns the reverse cumulative (upper-tail, survival)
noncentral F distribution with n1 numerator and n2
denominator degrees of freedom and noncentrality
parameter L.
returns 1 if f < 0.
nFtail() is computed using nibeta() based on the
relationship between the noncentral beta and F
distributions. See Johnson, Kotz, and Balakrishnan
(1995) for more details.
invnFtail(n1,n2,L,p)
Domain n1: 1e-323 to 8e+307 (may be nonintegral)
Domain n2: 1e-323 to 8e+307 (may be nonintegral)
Domain L: 0 to 1,000
Domain p: 0 to 1
Range: 0 to 8e+307
Description: returns the inverse reverse cumulative (upper-tail,
survival) noncentral F distribution: if
nFtail(n1,n2,L,f) = p, then invnFtail(n1,n2,L,p) =
x.
Gamma distribution
gammap(a,x)
Domain a: 1e-10 to 1e+17
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: 0 to 1
Description: returns the cumulative gamma distribution with shape
parameter a.
returns 0 if x < 0.
The cumulative Poisson (the probability of observing k
or fewer events if the expected is x) can be evaluated
as 1-gammap(k+1,x). The reverse cumulative (the
probability of observing k or more events) can be
evaluated as gammap(k,x).
gammap() is also known as the incomplete gamma function
(ratio).
Probabilities for the three-parameter gamma distribution
(see gammaden() can be calculated by shifting and
scaling x; i.e., gammap(a,(x - g)/b).
gammaden(a,b,g,x)
Domain a: 1e-323 to 8e+307
Domain b: 1e-323 to 8e+307
Domain g: -8e+307 to 8e+307
Domain x: -8e+307 to 8e+307
Interesting domain is x > g
Range: 0 to 8e+307
Description: returns the probability density function of the gamma
distribution, where a is the shape parameter, b is
the scale parameter, and g is the location
parameter.
returns 0 if x < g.
gammaptail(a,x)
Domain a: 1e-10 to 1e+17
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: 0 to 1
Description: returns the reverse (upper-tail, survival) cumulative
gamma distribution with shape parameter a.
returns 1 if x < 0.
gammaptail() is also known as the complement to the
incomplete gamma function (ratio).
invgammap(a,p)
Domain a: 1e-10 to 1e+17
Domain p: 0 to 1
Range: 0 to 8e+307
Description: returns the inverse cumulative gamma distribution: if
gammap(a,x) = p, then invgammap(a,p) = x.
invgammaptail(a,p)
Domain a: 1e-10 to 1e+17
Domain p: 0 to 1
Range: 0 to 8e+307
Description: returns the inverse reverse cumulative (upper-tail,
survival) gamma distribution: if gammaptail(a,x) =
p, then invgammaptail(a,p) = x.
dgammapda(a,x)
Domain a: 1e-7 to 1e+17
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: -16 to 0
Description: returns the partial derivative of the cumulative gamma
distribution gammap(a,x) with respect to a, for a >
0.
returns 0 if x < 0.
dgammapdada(a,x)
Domain a: 1e-7 to 1e+17
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: -0.02 to 4.77e+5
Description: returns the 2nd partial derivative of the cumulative
gamma distribution gammap(a,x) with respect to a,
for a > 0.
returns 0 if x < 0.
dgammapdadx(a,x)
Domain a: 1e-7 to 1e+17
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: -0.04 to 8e+307
Description: returns the 2nd partial derivative of the cumulative
gamma distribution gammap(a,x) with respect to a and
x, for a > 0.
returns 0 if x < 0.
dgammapdx(a,x)
Domain a: 1e-10 to 1e+17
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: 0 to 8e+307
Description: returns the partial derivative of the cumulative gamma
distribution gammap(a,x) with respect to x, for a >
0.
returns 0 if x < 0.
dgammapdxdx(a,x)
Domain a: 1e-10 to 1e+17
Domain x: -8e+307 to 8e+307
Interesting domain is x > 0
Range: 0 to 1e+40
Description: returns the 2nd partial derivative of the cumulative
gamma distribution gammap(a,x) with respect to x,
for a > 0.
returns 0 if x < 0.
Hypergeometric distribution
hypergeometric(N,K,n,k)
Domain N: 2 to 1e+5
Domain K: 1 to N-1
Domain n: 1 to N-1
Domain k: max(0,n-N+K) to min(K,n)
Range: 0 to 1
Description: returns the cumulative probability of the hypergeometric
distribution. N is the population size, K is the
number of elements in the population that have the
attribute of interest, and n is the sample size.
Returned is the probability of observing k or fewer
elements from a sample of size n that have the
attribute of interest.
hypergeometricp(N,K,n,k)
Domain N: 2 to 1e+5
Domain K: 1 to N-1
Domain n: 1 to N-1
Domain k: max(0,n-N+K) to min(K,n)
Range: 0 to 1 (right exclusive)
Description: returns the hypergeometric probability of k successes
(where success is obtaining an element with the
attribute of interest) out of a sample of size n,
from a population of size N containing K elements
that have the attribute of interest.
Negative binomial distribution
nbinomial(n,k,p)
Domain n: 1e-10 to 1e+17 (can be nonintegral)
Domain k: 0 to 2^53-1
Domain p: 0 to 1 (left exclusive)
Range: 0 to 1
Description: returns the cumulative probability of the negative
binomial distribution. n can be nonintegral. When
n is an integer, nbinomial() returns the probability
of observing k or fewer failures before the nth
success, when the probability of a success on one
trial is p.
The negative binomial distribution function is evaluated
using the ibeta() function.
nbinomialp(n,k,p)
Domain n: 1e-10 to 1e+6 (can be nonintegral)
Domain k: 0 to 1e+10
Domain p: 0 to 1 (left exclusive)
Range: 0 to 1
Description: returns the negative binomial probability. When n is an
integer, nbinomialp() returns the probability of
observing exactly floor(k) failures before the nth
success when the probability of a success on one
trial is p.
nbinomialtail(n,k,p)
Domain n: 1e-10 to 1e+17 (can be nonintegral)
Domain k: 0 to 2^53-1
Domain p: 0 to 1 (left exclusive)
Range: 0 to 1
Description: returns the reverse cumulative probability of the
negative binomial distribution. When n is an
integer, nbinomialtail() returns the probability of
observing k or more failures before the nth success,
when the probability of a success on one trial is p.
The reverse negative binomial distribution function is
evaluated using the ibetatail() function.
invnbinomial(n,k,q)
Domain n: 1e-10 to 1e+17 (can be nonintegral)
Domain k: 0 to 2^53-1
Domain q: 0 to 1 (exclusive)
Range: 0 to 1 (exclusive)
Description: returns the value of the negative binomial parameter, p,
such that q = nbinomial(n,k,p).
invnbinomial() is evaluated using invibeta().
invnbinomialtail(n,k,q)
Domain n: 1e-10 to 1e+17 (can be nonintegral)
Domain k: 1 to 2^53-1
Domain q: 0 to 1 (exclusive)
Range: 0 to 1 (exclusive)
Description: returns the value of the negative binomial parameter, p,
such that q = nbinomialtail(n,k,p).
invnbinomialtail() is evaluated using invibetatail().
Normal (Gaussian), log of the normal, and binormal distributions
binormal(h,k,r)
Domain h: -8e+307 to 8e+307
Domain k: -8e+307 to 8e+307
Domain r: -1 to 1
Range: 0 to 1
Description: returns the joint cumulative distribution of the
bivariate normal with correlation r; cumulative over
(-inf,h] x (-inf,k].
normal(z)
Domain: -8e+307 to 8e+307
Range: 0 to 1
Description: returns the cumulative standard normal distribution.
normalden(z)
Domain: -8e+307 to 8e+307
Range: 0 to 0.39894 ...
Description: returns the standard normal density.
normalden(z,s)
Domain z: -8e+307 to 8e+307
Domain s: 1e-308 to 8e+307
Range: 0 to 8e+307
Description: returns the rescaled standard normal density.
normalden(z,1) = normalden(x) and normalden(z,s) =
normalden(z)/s.
normalden(x,m,s)
Domain x: -8e+307 to 8e_307
Domain m: -8e+307 to 8e_307
Domain s: 1e-308 to 8e+307
Range: 0 to 8e+307
Description: returns the normal density with mean m and standard
deviation s.
normalden(x,0,1) = normalden(x)
normalden(x,m,s) = normalden((x-m)/s)/s
invnormal(p)
Domain: 1e-323 to 1 - 2^(-53)
Range: -39 to 8.2095362
Description: returns the inverse cumulative standard normal
distribution: if normal(z) = p, then invnormal(p) =
z.
lnnormal(z)
Domain: -1e+99 to 8e+307
Range: -5e+197 to 0
Description: returns the natural logarithm of the cumulative standard
normal distribution.
lnnormalden(z)
Domain: -1e+154 to 1e+154
Range: -5e+307 to -.91893853 = lnnormalden(0)
Description: returns the natural logarithm of the standard normal
density.
lnnormalden(z,s)
Domain z: -1e+154 to 1e+154
Domain s: 1e-323 to 8e+307
Range: -5e+307 to 742.82799
Description: returns the natural logarithm of the rescaled standard
normal density.
lnnormalden(z,1) = lnnormalden(z)
lnnormalden(z,s) = lnnormalden(z)/ln(s)
lnnormalden(x,m,s)
Domain x: -8e+307 to 8e+307
Domain m: -8e+307 to 8e+307
Domain s: 1e-323 to 8e+307
Range: 1e-323 to 8e+307
Description: returns the natural logarithm of the normal density with
mean m and standard deviation s:
lnnormalden(x,0,1) = lnnormalden(x) and
lnnormalden(x,m,s) = lnnormalden((x-m)/s) - ln(s).
Poisson distribution
poisson(m,k)
Domain m: 1e-10 to 2^53-1
Domain k: 0 to 2^53-1
Range: 0 to 1
Description: returns the probability of observing floor(k) or fewer
outcomes that are distributed as Poisson with mean
m.
The Poisson distribution function is evaluated using the
gammaptail() function.
poissonp(m,k)
Domain m: 1e-10 to 1e+8
Domain k: 0 to 1e+9
Range: 0 to 1
Description: returns the probability of observing floor(k) outcomes
that are distributed as Poisson with mean m.
The Poisson probability function is evaluated using the
gammaden() function.
poissontail(m,k)
Domain m: 1e-10 to 2^53-1
Domain k: 0 to 2^53-1
Range: 0 to 1
Description: returns the probability of observing floor(k) or more
outcomes that are distributed as Poisson with mean
m.
The reverse cumulative Poisson distribution function is
evaluated using the gammap() function.
invpoisson(k,p)
Domain k: 0 to 2^53-1
Domain p: 0 to 1 (exclusive)
Range: 0 to 2^53
Description: returns the Poisson mean such that the cumulative
Poisson distribution evaluated at k is p: if
poisson(m,k) = p, then invpoisson(k,p) = m.
The inverse Poisson distribution function is evaluated
using the invgammaptail() function.
invpoissontail(k,q)
Domain k: 0 to 2^53-1
Domain q: 0 to 1 (exclusive)
Range: 0 to 2^53
Description: returns the Poisson mean such that the reverse
cumulative Poisson distribution evaluated at k is q:
if poissontail(m,k) = q, then invpoissontail(k,q) =
m.
The inverse of the reverse cumulative Poisson
distribution function is evaluated using the invgammap()
function.
Student's t distribution
tden(n,t)
Domain n: 1e-323 to 8e+307
Domain t: -8e+307 to 8e+307
Range: 0 to 0.39894 ...
Description: returns the probability density function of Student's t
distribution.
ttail(n,t)
Domain n: 2e-10 to 2e+17
Domain t: -8e+307 to 8e+307
Range: 0 to 1
Description: returns the reverse cumulative (upper-tail, survival)
Student's t distribution; it returns the probability
T > t.
invttail(n,p)
Domain n: 2e-10 to 2e+17 (may be nonintegral)
Domain p: 0 to 1
Range: -8e+307 to 8e+307
Description: returns the inverse reverse cumulative (upper-tail,
survival) Student's t distribution: if ttail(n,t) =
p, then invttail(n,p) = t.
References
Haynam, G. E., Z. Govindarajulu, and F. C. Leone. 1970. Tables of the
cumulative noncentral chi-square distribution. In Vol. 1 of Selected
Tables in Mathematical Statistics, ed. H. L. Harter and D. B. Owen,
1-78. Providence, RI: American Mathematical Society.
Johnson, N. L., S. Kotz, and N. Balakrishnan. 1995. Continuous
Univariate Distributions, Vol. 2. 2nd ed. New York: Wiley.
Also see
Manual: [D] functions
Help: [D] egen
