# Mario Becerra

## Pi estimation using simulation

There are many ways to estimate the value of $$\pi$$, some more efficient than others. One of them is using Monte Carlo simulation. The idea is that if you have a circle of radius 1, it will have an area of $$\pi$$. You can fit a circle within a square of area 4, such that they both have a centroid in the origin, like shown in the following image.

Then, if one simulates $$N$$ points in the square, counts how many were inside the circle ($$N_{in}$$) and computes $$\frac{N_{in}}{N}$$, one gets a number close to $$\frac{\pi}{4}$$. And the bigger $$N$$ gets, the closer one is to the real value of $$\pi$$. So, to estimate $$\pi$$, one computes $$4\frac{N_{in}}{N}$$. An example of the simulated points can be seen in the following figure (done in R).

tibble(x = runif(10000, -1, 1),
y = runif(10000, -1, 1)) %>%
mutate(inside = x^2 + y^2 < 1) %>%
ggplot() +
geom_point(aes(x, y, color = inside), size = 0.5) +
coord_equal() +
theme_bw()