The idea behind this post is to collect various numerical methods to simulate discrete probability problems.

Expectation of a uniform variable.

Question: what is the expectation if one square a number that is picked at random out of a hat (with replacement) that contains the numbers 1 to 100.

expec <- mean(sample(1:100, size = 1000000, replace = TRUE)^2)
print(expec)

[1] 3384.936

The calculated expectation should be: \[\sum_{x=1}^{100} x^2 P(X=x) = \sum_{x=1}^{100} x^2 \frac{1}{n} = \frac{101 \cdot 201}{6} = 3383.5\]

We could connect this to the Jensen’s inequality (as we are dealing with a convex function) and show that indeed the expectation of the function is greater than the function of the expectation: \(\mathbb{E}[f(X)] \geq f(\mathbb{E}[X])\)

exp_square <- mean(sample(1:100, 1000000, replace = TRUE)^2)
square_exp <- (mean(sample(1:100, 1000000, replace = TRUE)))^2

print(exp_square)

[1] 3389.425

print(square_exp)

[1] 2551.491