Gaussian random variable and CLT

To generate two independent Gaussian random variables

$x$ and

$y$ we use these equation

$$x=\sigma \sqrt{-2ln{\xi }_1}\cos (2\pi {\xi }_2),y=\sigma \sqrt{-2ln{\xi }_1}\sin (2\pi {\xi }_2)$$
This figure show total number

$N={10}^5$ Gaussian random numbers.

We can plot the semi-log to again verified the result.

For generating two independent Gaussian variables

$x_1$ and

$x_2$ to form a distribution

$y=x_1+x_2$, we use same approach above.I then use KDE (kernel density estimation) to fit two Gaussian distribution.

Interesting happened when I generate

$n$independent uniform random numbers

${\xi }_1$,

${\xi }_2$, …,

${\xi }_n$, then I measure the probability of the random variable

$y={\sigma }_{k=1}^n{\xi }_k$. As you can see when

$n=1$ it is just a uniform random distribution,
however when

$n=2$ or even larger number, the distribution looks gradually like Gaussian distribution. At

$n=20$, it looks like a pretty decent Gaussian distribution which I use theoretical result which states that

$$\mu ={\displaystyle \frac{1}{2}}{\sigma }_n^2={\displaystyle \frac{1}{12n}}$$

I then use totally different random distribution consist of one-third of exponential distribution, another one-third of uniform distribution and one-third of Gaussian distribution. The parameter are given below.

$$P_1(x)=0.2e^{-0.2x}{P}_2(x)=1P_3(x)={\displaystyle \frac{1}{\sqrt{2\pi \times 0.2}}}{e}^{-{\displaystyle \frac{(x-\mu {)}^2}{2\times 0.2}}}$$
As you can see it become Gaussian again when n is larger (roughly at

$n=10$).

Code

Gaurand

program main
    implicit none
    double precision, external :: gaurand
    integer, parameter :: n=100000
    double precision :: x(n), sig=dsqrt(.2d0)
    integer :: i
    do i = 1, n
        x(i) = gaurand(sig)
    end do

    open(66, file='a.dat', status='unknown')
    do i = 1, n
        write(66, *) x(i)
    end do
end program main

function gaurand(sig)
    implicit none
    double precision, parameter :: tpi=8.d0*atan(1.d0)
    double precision :: a, b, gaurand, sig
    call random_seed()
    call random_number(a)
    call random_seed()
    call random_number(b)
    gaurand = sig * dsqrt(-2.d0 * log(a)) * dcos(tpi * b)
end function

CLT

program clt
    implicit none
    integer, parameter :: n=1000, m=50
    double precision :: x(n, m), y(n)
    integer :: i, j
    open(66, file='c50.dat', status='unknown')
    call random_seed()
    do i = 1, n
        call random_number(x)
    end do

    y = 0.d0
    do j = 1, m
        y(:) = y(:) + x(:, j)
    end do
    y = y / m

    do i = 1, n
        write(66, *) y(i)
    end do

end program clt

program clt
    implicit none
    double precision, external :: gaurand
    integer, parameter :: n=10000, m=3
    double precision :: x1(n, m), x2(n, m), x3(n, m), y(n)
    integer :: i, j
    double precision :: a = .2d0
    open(66, file='d3.dat', status='unknown')
    call random_seed()

    call random_number(x1)
    x1 = - 1.d0 / a * dlog(x1)

    call random_number(x2)

    do j = 1, m
        do i = 1, n
            x3(i, j) = gaurand(dsqrt(.2d0))
        end do
    end do

    y = 0.d0
    do j = 1, m
        y(:) = y(:) + x1(:, j) + x2(:, j)+ x3(:, j)
    end do
    y = y / m

    do i = 1, n
        write(66, *) y(i)
    end do

end program clt

function gaurand(sig)
    implicit none
    double precision, parameter :: tpi=8.d0*atan(1.d0)
    double precision :: a, b, gaurand, sig
    call random_seed()
    call random_number(a)
    call random_seed()
    call random_number(b)
    gaurand = sig * dsqrt(-2.d0 * log(a)) * dcos(tpi * b)
end function