How to estimate the value of π…(part 2)

In my last blog entry, I had ended by introducing the idea why a Monte-Carlo simulation that estimates a $\pi$ without estimating the error involved in the simulation gives an incomplete picture of the simulation. The Monte Carlo Standard Error (MCSE) is a measure of the accuracy of the simulation. The lower the MCSE, the more accurate the simulation.

Let $\widehat{\pi}$ be the Monte Carlo-estimated value of $pi$ and let $\widehat{\sigma}$ be the corresponding standard deviation of the simulation. The MCSE of the simulation would be $\frac{\widehat{\sigma}}{n}$ where n would be the size of the generated scenarios (I belabor this point but we generated $2n$ numbers so we could get $n$ -pairs of numbers and thus we have $n$ scenarios)

So, let us tweak the previous program to change the function estimate_pi()

def estimate_pi(n=1000000):: nums = 2 * np.random.random_sample((n, 2)) - 1.0 n = len(nums) insides = np.array([1 if inside_circle(x, y) else 0 for x, y in nums]) mean = np.mean(insides) sd = np.std(insides) return 4 * mean, 4 * sd/sqrt(n)