Ggtkuk

I am trying to 'simulate' rnorm() using only runif().

I don't know if I should do:

sqrt(-2*log(U1))*cos(U2)

or

sqrt(-2*log(U1))*sin(U2)

Where U1 is a runif(0,1) and U2 runif(0,6.28)

I do not know if I should do it using cos or sin, or is it that I need to sample from one and the other consecutively? What is the mathematical logic behind it?

It looks like you are trying to use the Box-Muller transform.

The method is to use $U_1, U_2$ which are independently distribution Uniform(0,1). Then

$Z_0 = sqrt-2 ln(U_1)cos(2 pi U_2) $

and

$Z_1 = sqrt-2 ln(U_1) sin(2 pi U_2) $

are a pair of independent $N(0,1)$.

The derivation comes from using polar coordinates, but to be honest the exact steps escape me at the moment (see link if you are interested).

Let $N_1,N_2$ be independent $N(0,1)$ random variates. $(N_1,N_2)$ defines a point in Cartesian coordinates. We transform to polar by

$$ N_1 = R cos theta$$
$$ N_2 = R sin theta$$

This transformation is one to one and has continuous derivatives. So we can derive the joint distribution $f_R,theta(r,theta)$ by using

$$f_R,theta(r,theta) = f_N_1,N_2(n_1,n_2)|J^-1|$$

where $J$ is the Jacobian of the transformation and

$$ J^-1(r,theta )=
beginbmatrixdfrac partial n_1partial rdfrac partial n_1partial theta \dfrac partial n_2partial rdfrac partial n_2partial theta endbmatrix
= beginbmatrixcos theta & -rsin theta \sin theta & rcos theta endbmatrix$$

Hence $$ |J^-1| = r cos^2 theta + r sin^2 theta = r $$

and

$$ f_R,theta(r,theta) = r frac12pi e^-frac12(n_1^2+n_2^2) = boxedfracr2 pie^-frac12(r^2) qquad 0 leq theta leq 2 pi, 0 leq r leq infty $$

And we see $theta sim unif[0,2pi]$ (since $f_Theta(theta) = frac12pi$ ) and $R^2 sim exp(1/2)$

Hence, to simulate $Theta$ we simply take $2 pi U_2$ where $$U_2 sim unif[0,1]$$ and to simulate $R$ we can take $$-2ln U_1$$ where $U_1 sim unif[0,1]$. ( to see why find the density of $X=-ln U, U sim unif[0,1]$).

So, Box and Muller simply inverted $N_1= R cos theta$, $N_2= R sin theta$ and moved from $(R,Theta)$ to $(N_1,N_2)$ by simulating $Theta$ from $2 pi U_2$, and an independent $R$ from $ sqrt- 2 ln U_1$

explicitly

$$Z_0 = sqrt-2 ln(U_1)cos(2 pi U_2)$$

$$Z_1 = sqrt-2 ln(U_1) sin(2 pi U_2)$$

And that is that is the mathematical logic behind it.

p.s. As I don't know whether it is clear through the mathematical justification (I believe it is, but you may have been lost in details), note you need a pair of drawings from the uniform to get a pair of normals