PaC Exercise 2.1

import numpy as np

Exercise 2.1

Suppose we roll a fair k-sided die with the numbers 1 through k on the die’s faces. If X is the number that appears, what is E(X)?

Solution:

$$ E(X) = \sum_{i=1}^K i \cdot P(X=i) = \sum_{i=1}^K i \frac 1 K = \frac 1 K \frac{K(K+1)}{2} = \frac{K+1}{2}, $$

by using the Gauss’ infamous summation formula.

This is consistent with a 6 sided die having expectation value of 7/2.

We can also approximate the expectation value of a k = 117-sided die, which should be 118/2 = 59, by sampling as follows.

num_samples = 100000
k = 117

np.random.randint(low=1, high=k+1, size=num_samples).mean()

59.03891