Histogram Plot Gaussian Distribution

Question

I have to construct functions to obtain random numbers from a Gaussian Distribution with mean $\mu$ and variance $\sigma^2$ by using box-muller method and testing the function by sampling from a Gaussian with $\mu=10$ and $\sigma^2=5$. I have to plot the histograms together from a sufficient number of samples with the given distribution function. However, I cannot use NormalDistribution[$\mu$,$\sigma$].

Any ideas on how to do this would be highly appreciated.

For the box-muller method I managed to get this code:

u1 = RandomReal[{0, 1}, 20];
u2 = RandomReal[{0, 1}, 20];
r = Sqrt[-2*Log[u1]];
v = 2*Pi*u2;
x = r*Cos[v]
y = r*Sin[v]

Answer 1

With only slight modification to your equations for the box-muller method:

n = 1000;
mu = 10;
sigma = Sqrt[5];
SeedRandom[1];
u1 = RandomReal[{0, 1}, n];
u2 = RandomReal[{0, 1}, n];
r = sigma Sqrt[-2*Log[u1]];
v = 2*Pi*u2;
x = r*Cos[v] + mu
y = r*Sin[v] + mu
StandardDeviation /@ {x, y}^2
Mean /@ {x, y}

For n = 1000 samples and with RandomSeed[1], the sigma squared of x and y respectively is {5.08208, 4.83115} and the mean of x and y respectively is {9.95581, 10.0357}

We can make a histogram and compare it to the normal distribution like this:

Show[Plot[
  1/(sigma Sqrt[2 Pi]) Exp[-1/2 ((x - mu)/sigma)^2], {x, 0, 20}, 
  PlotLegends -> {"Normal Distribution"}],
 Histogram[{x, y}, Automatic, "PDF", ChartLegends -> {"X", "Y"}]]

With n = 1,000,000 samples and changing the bin width specification from "Automatic" to {.1} we get a smoother graph.