rvs_exp - exponential distribution random variatesExperimental
An exponential distribution is the distribution of time between events in a Poisson point process.
The inverse scale parameter lambda specifies the average time between events (), also called the rate of events. The location loc specifies the value by which the distribution is shifted.
Without argument, the function returns a random sample from the unshifted standard exponential distribution or .
With a single argument of type real, the function returns a random sample from the exponential distribution .
For complex arguments, the real and imaginary parts are sampled independently of each other.
With one argument of type real and one argument of type integer, the function returns a rank-1 array of exponentially distributed random variates for (E(\lambda=\text{lambda})).
With two arguments of type real, the function returns a random sample from the exponential distribution .
For complex arguments, the real and imaginary parts are sampled independently of each other.
With two arguments of type real and one argument of type integer, the function returns a rank-1 array of exponentially distributed random variates for .
Note
The algorithm used for generating exponential random variates is fundamentally limited to double precision.1
result = rvs_exp ([loc, scale] [[, array_size]])
or
result = rvs_exp ([lambda] [[, array_size]])
Elemental function
lambda: optional argument has intent(in) and is a scalar of type real or complex.
If lambda is real, its value must be positive. If lambda is complex, both the real and imaginary components must be positive.
loc: optional argument has intent(in) and is a scalar of type real or complex.
scale: optional argument has intent(in) and is a positive scalar of type real or complex.
If scale is real, its value must be positive. If scale is complex, both the real and imaginary components must be positive.
array_size: optional argument has intent(in) and is a scalar of type integer with default kind.
If lambda is passed, the result is a scalar or rank-1 array with a size of array_size, and the same type as lambda.
If lambda is non-positive, the result is NaN.
If loc and scale are passed, the result is a scalar or rank-1 array with a size of array_size, and the same type as scale.
If scale is non-positive, the result is NaN.
program example_exponential_rvs
use stdlib_random, only: random_seed
use stdlib_stats_distribution_exponential, only: rexp => rvs_exp
implicit none
complex :: cloc, cscale
integer :: seed_put, seed_get
seed_put = 1234567
call random_seed(seed_put, seed_get)
! single standard exponential random variate
print *, rexp()
! 0.358690143
! exponential random variate with loc=0 and scale=0.5 (lambda=2)
print *, rexp(0.0, 0.5)
! 0.122672431
! exponential random variate with lambda=2
print *, rexp(2.0)
! 0.204114929
! exponential random variate with loc=0.6 and scale=0.2 (lambda=5)
print *, rexp(0.6, 0.2)
! 0.681645989
! an array of 10 variates with loc=0.0 and scale=3.0 (lambda=1/3)
print *, rexp(0.0, 3.0, 10)
! 1.36567295 2.62772131 0.362352759 5.47133636 2.13591909
! 0.410784155 5.83882189 6.71128035 1.31730068 1.90963650
! single complex exponential random variate with real part of scale=0.5 (lambda=2.0);
! imagainary part of scale=1.6 (lambda=0.625)
cloc = (0.0, 0.0)
cscale = (0.5, 1.6)
print *, rexp(cloc, cscale)
! (0.426896989,2.56968451)
end program example_exponential_rvs
pdf_exp - exponential distribution probability density functionExperimental
The probability density function (pdf) of the single real variable exponential distribution is:
For a complex variable with independent real and imaginary parts, the joint probability density function is the product of the corresponding real and imaginary marginal pdfs:2
Instead of of the inverse scale parameter lambda, it is possible to pass loc and scale, where and loc specifies the value by which the distribution is shifted.
result = pdf_exp (x, loc, scale)
or
result = pdf_exp (x, lambda)
Elemental function
x: has intent(in) and is a scalar of type real or complex.
lambda: has intent(in) and is a scalar of type real or complex.
If lambda is real, its value must be positive. If lambda is complex, both the real and imaginary components must be positive.
loc: has intent(in) and is a scalar of type real or complex.
scale: has intent(in) and is a positive scalar of type real or complex.
If scale is real, its value must be positive. If scale is complex, both the real and imaginary components must be positive.
All arguments must have the same type.
The result is a scalar or an array, with a shape conformable to the arguments, and the same type as the input arguments. If non-positive lambda or scale, the result is NaN.
program example_exponential_pdf
use stdlib_random, only: random_seed
use stdlib_stats_distribution_exponential, only: exp_pdf => pdf_exp, &
rexp => rvs_exp
implicit none
real, dimension(2, 3, 4) :: x, loc, scale
real :: xsum
complex :: cloc, cscale
integer :: seed_put, seed_get, i
seed_put = 1234567
call random_seed(seed_put, seed_get)
! probability density at x=1.0 with loc=0 and scale=1.0
print *, exp_pdf(1.0, 0.0, 1.0)
! 0.367879450
! probability density at x=1.0 with lambda=1.0
print *, exp_pdf(1.0, 1.0)
! 0.367879450
! probability density at x=2.0 with lambda=2.0
print *, exp_pdf(2.0, 2.0)
! 3.66312787E-02
! probability density at x=2.0 with loc=0.0 and scale=0.5 (lambda=2.0)
print *, exp_pdf(2.0, 0.0, 0.5)
! 3.66312787E-02
! probability density at x=1.5 with loc=0.5 and scale=0.5 (lambda=2.0)
print *, exp_pdf(2.5, 0.5, 0.5)
! 3.66312787E-02
! probability density at x=2.0 with loc=0.0 and scale=-1.0 (out of range)
print *, exp_pdf(2.0, 0.0, -1.0)
! NaN
! standard exponential random variates array
x = reshape(rexp(0.0, 2.0, 24), [2, 3, 4])
! a rank-3 exponential probability density
loc(:, :, :) = 0.0
scale(:, :, :) = 2.0
print *, exp_pdf(x, loc, scale)
! 0.349295378 0.332413018 0.470253497 0.443498343 0.317152828
! 0.208242029 0.443112582 8.07073265E-02 0.245337561 0.436016470
! 7.14025944E-02 5.33841923E-02 0.322308093 0.264558554 0.212898195
! 0.100339092 0.226891592 0.444002301 9.91026312E-02 3.87373678E-02
! 3.11400592E-02 0.349431813 0.482774824 0.432669312
! probability density array where scale<=0.0 for certain elements (loc = 0.0)
print *, exp_pdf([1.0, 1.0, 1.0], [0.0, 0.0, 0.0], [1.0, 0.0, -1.0])
! 0.367879450 NaN NaN
! `pdf_exp` is pure and, thus, can be called concurrently
xsum = 0.0
do concurrent (i=1:size(x,3))
xsum = xsum + sum(exp_pdf(x(:,:,i), loc(:,:,i), scale(:,:,i)))
end do
print *, xsum
! 6.45566940
! complex exponential probability density function at (1.5, 0.0, 1.0) with real part
! of scale=1.0 and imaginary part of scale=0.5
cloc = (0.0, 0.0)
cscale = (1.0, 0.5)
print *, exp_pdf((1.5, 1.0), cloc, cscale)
! 6.03947677E-02
! As above, but with scale%re < 0
cloc = (0.0, 0.0)
cscale = (-1.0, 2.0)
print *, exp_pdf((1.5, 1.0), cloc, cscale)
! NaN
end program example_exponential_pdf
cdf_exp - exponential cumulative distribution functionExperimental
Cumulative distribution function (cdf) of the single real variable exponential distribution:
For a complex variable with independent real and imaginary parts, the joint cumulative distribution function is the product of corresponding real and imaginary marginal cdfs:2
Alternative to the inverse scale parameter lambda, it is possible to pass loc and scale, where and loc specifies the value by which the distribution is shifted.
result = cdf_exp (x, loc, scale)
or
result = cdf_exp (x, lambda)
Elemental function
x: has intent(in) and is a scalar of type real or complex.
lambda: has intent(in) and is a scalar of type real or complex.
If lambda is real, its value must be positive. If lambda is complex, both the real and imaginary components must be positive.
loc: has intent(in) and is a scalar of type real or complex.
scale: has intent(in) and is a positive scalar of type real or complex.
If scale is real, its value must be positive. If scale is complex, both the real and imaginary components must be positive.
All arguments must have the same type.
The result is a scalar or an array, with a shape conformable to the arguments, and the same type as the input arguments. With non-positive lambda or scale, the result is NaN.
program example_exponential_cdf
use stdlib_random, only: random_seed
use stdlib_stats_distribution_exponential, only: exp_cdf => cdf_exp, &
rexp => rvs_exp
implicit none
real, dimension(2, 3, 4) :: x, loc, scale
real :: xsum
complex :: cloc, cscale
integer :: seed_put, seed_get, i
seed_put = 1234567
call random_seed(seed_put, seed_get)
! standard exponential cumulative distribution at x=1.0 with loc=0.0, scale=1.0
print *, exp_cdf(1.0, 0.0, 1.0)
! 0.632120550
! standard exponential cumulative distribution at x=1.0 with lambda=1.0
print *, exp_cdf(1.0, 1.0)
! 0.632120550
! cumulative distribution at x=2.0 with lambda=2
print *, exp_cdf(2.0, 2.0)
! 0.981684387
! cumulative distribution at x=2.0 with loc=0.0 and scale=0.5 (equivalent of lambda=2)
print *, exp_cdf(2.0, 0.0, 0.5)
! 0.981684387
! cumulative distribution at x=2.5 with loc=0.5 and scale=0.5 (equivalent of lambda=2)
print *, exp_cdf(2.5, 0.5, 0.5)
! 0.981684387
! cumulative distribution at x=2.0 with loc=0.0 and scale=-1.0 (out of range)
print *, exp_cdf(2.0, 0.0, -1.0)
! NaN
! cumulative distribution at x=0.5 with loc=1.0 and scale=1.0, putting x below the minimum
print *, exp_cdf(0.5, 1.0, 1.0)
! 0.00000000
! standard exponential random variates array
x = reshape(rexp(0.0, 2.0, 24), [2, 3, 4])
! a rank-3 exponential cumulative distribution
loc(:, :, :) = 0.0
scale(:, :, :) = 2.0
print *, exp_cdf(x, loc, scale)
! 0.301409245 0.335173965 5.94930053E-02 0.113003314
! 0.365694344 0.583515942 0.113774836 0.838585377
! 0.509324908 0.127967060 0.857194781 0.893231630
! 0.355383813 0.470882893 0.574203610 0.799321830
! 0.546216846 0.111995399 0.801794767 0.922525287
! 0.937719882 0.301136374 3.44503522E-02 0.134661376
! cumulative distribution array where scale<=0.0 for certain elements
print *, exp_cdf([1.0, 1.0, 1.0], [0.0, 0.0, 0.0], [1.0, 0.0, -1.0])
! 0.632120550 NaN NaN
! `cdf_exp` is pure and, thus, can be called concurrently
xsum = 0.0
do concurrent (i=1:size(x,3))
xsum = xsum + sum(exp_cdf(x(:,:,i), loc(:,:,i), scale(:,:,i)))
end do
print *, xsum
! 11.0886612
! complex exponential cumulative distribution at (0.5, 0.0, 2) with real part of
! scale=2 and imaginary part of scale=1.0
cloc = (0.0, 0.0)
cscale = (2, 1.0)
print *, exp_cdf((0.5, 0.5), cloc, cscale)
! 8.70351046E-02
! As above, but with scale%im < 0
cloc = (0.0, 0.0)
cscale = (1.0, -2.0)
print *, exp_cdf((1.5, 1.0), cloc, cscale)
! NaN
end program example_exponential_cdf
Marsaglia, George, and Wai Wan Tsang. "The ziggurat method for generating random variables." Journal of statistical software 5 (2000): 1-7. ↩
Miller, Scott, and Donald Childers. Probability and random processes: With applications to signal processing and communications. Academic Press, 2012 (p. 197). ↩↩