# Statistical Distributions -- Exponential Distribution Module

## rvs_exp - exponential distribution random variates

Experimental

### Description

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.

Without argument the function returns a random sample from the standard exponential distribution E(1) with lambda = 1.

With a single argument, the function returns a random sample from the exponential distribution E(lambda). For complex arguments, the real and imaginary parts are sampled independently of each other.

With two arguments the function returns a rank one array of exponentially distributed random variates.

Note: the algorithm used for generating exponetial random variates is fundamentally limited to double precision. Ref.: Marsaglia, G. & Tsang, W.W. (2000) The ziggurat method for generating random variables', J. Statist. Software, v5(8).

### Syntax

result = rvs_exp([lambda] [[, array_size]])

Function

### Arguments

lambda: optional argument has intent(in) and is a scalar of type real or complex. The value of lambda has to be non-negative.

array_size: optional argument has intent(in) and is a scalar of type integer with default kind.

### Return value

The result is a scalar or rank one array with a size of array_size, and has the same type of lambda.

### Example

program demo_exponential_rvs
use stdlib_random, only : random_seed
use stdlib_stats_distribution_exponential, only: rexp => rvs_exp

implicit none
real ::  a(2,3,4)
complex :: scale
integer :: seed_put, seed_get

seed_put = 1234567
call random_seed(seed_put, seed_get)

print *, rexp( )         !single standard exponential random variate

! 0.358690143

print *, rexp(2.0)       !exponential random variate with lambda=2.0

! 0.816459715

print *, rexp(0.3, 10)   !an array of 10 variates with lambda=0.3

!  1.84008647E-02  3.59742008E-02  0.136567295  0.262772143  3.62352766E-02
!  0.547133625  0.213591918  4.10784185E-02  0.583882213  0.671128035

scale = (2.0, 0.7)
print *, rexp(scale)
!single complex exponential random variate with real part of lambda=2.0;
!imagainary part of lambda=0.7

! (1.41435969,4.081114382E-02)

end program demo_exponential_rvs


## pdf_exp - exponential distribution probability density function

Experimental

### Description

The probability density function (pdf) of the single real variable exponential distribution:

For a complex variable (x + y i) with independent real x and imaginary y parts, the joint probability density function is the product of the corresponding marginal pdf of real and imaginary pdf (for more details, see "Probability and Random Processes with Applications to Signal Processing and Communications", 2nd ed., Scott L. Miller and Donald Childers, 2012, p.197):

### Syntax

result = pdf_exp(x, lambda)

### Class

Elemental function

### Arguments

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.

All arguments must have the same type.

### Return value

The result is a scalar or an array, with a shape conformable to arguments, and has the same type of input arguments.

### Example

program demo_exponential_pdf
use stdlib_random, only : random_seed
use stdlib_stats_distribution_exponential, only: exp_pdf => pdf_exp,     &
rexp => rvs_exp

implicit none
real :: x(2,3,4),a(2,3,4)
complex :: scale
integer :: seed_put, seed_get

seed_put = 1234567
call random_seed(seed_put, seed_get)

print *, exp_pdf(1.0,1.0) !a probability density at 1.0 in standard expon

! 0.367879450

print *, exp_pdf(2.0,2.0) !a probability density at 2.0 with lambda=2.0

! 3.66312787E-02

x = reshape(rexp(0.5, 24),[2,3,4]) ! standard expon random variates array
a(:,:,:) = 0.5
print *, exp_pdf(x, a)     ! a rank 3 standard expon probability density

!  0.457115263  0.451488823  0.492391467  0.485233188  0.446215510
!  0.401670188  0.485127628  0.316924453  0.418474048  0.483173639
!  0.307366133  0.285812140  0.448017836  0.426440030  0.403896868
!  0.334653258  0.410376132  0.485370994  0.333617479  0.263791025
!  0.249779820  0.457159877  0.495636940  0.482243657

scale = (1.0, 2.)
print *, exp_pdf((1.5,1.0), scale)
! a complex expon probability density function at (1.5,1.0) with real part
!of lambda=1.0 and imaginary part of lambda=2.0

! 6.03947677E-02

end program demo_exponential_pdf


## cdf_exp - exponential cumulative distribution function

Experimental

### Description

Cumulative distribution function (cdf) of the single real variable exponential distribution:

For a complex variable (x + y i) with independent real x and imaginary y parts, the joint cumulative distribution function is the product of corresponding marginal cdf of real and imaginary cdf (for more details, see "Probability and Random Processes with Applications to Signal Processing and Communications", 2nd ed., Scott L. Miller and Donald Childers, 2012, p.197):

### Syntax

result = cdf_exp(x, lambda)

### Class

Elemental function

### Arguments

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.

All arguments must have the same type.

### Return value

The result is a scalar or an array, with a shape conformable to arguments, and has the same type of input arguments.

### Example

program demo_exponential_cdf
use stdlib_random, only : random_seed
use stdlib_stats_distribution_exponential, only : exp_cdf => cdf_exp,    &
rexp => rvs_exp

implicit none
real :: x(2,3,4),a(2,3,4)
complex :: scale
integer :: seed_put, seed_get

seed_put = 1234567
call random_seed(seed_put, seed_get)

print *, exp_cdf(1.0, 1.0)  ! a standard exponential cumulative at 1.0

! 0.632120550

print *, exp_cdf(2.0, 2.0) ! a cumulative at 2.0 with lambda=2

! 0.981684387

x = reshape(rexp(0.5, 24),[2,3,4])
! standard exponential random variates array
a(:,:,:) = 0.5
print *, exp_cdf(x, a)  ! a rank 3 array of standard exponential cumulative

!  8.57694745E-02  9.70223546E-02  1.52170658E-02  2.95336246E-02
!  0.107568979  0.196659625  2.97447443E-02  0.366151094  0.163051903
!  3.36527228E-02  0.385267735  0.428375721  0.103964329  0.147119939
!  0.192206264  0.330693483  0.179247737  2.92580128E-02  0.332765043
!  0.472417951  0.500440359  8.56802464E-02  8.72612000E-03  3.55126858E-02

scale = (0.5,1.0)
print *, exp_cdf((0.5,0.5),scale)
!complex exponential cumulative distribution at (0.5,0.5) with real part of
!lambda=0.5 and imaginary part of lambda=1.0

! 8.70351046E-02

end program demo_exponential_cdf
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