import numpy as np

Our goals are to

• practice Python
• practice general expectations, namely

\[ \mathbb{E}[g(X)] = \sum_{x \in S} g(x) \cdot f(x) \]

All of this will only work for a specific class of distributions, namely ones which have a finite sample space. This won’t work for more general distributions, which we will get to in class soon.

def general_expectation(s, f, g, **kwargs):
    return np.sum(g(s, **kwargs) * f(s, **kwargs))

def uniform_density(x, a = 1, b = 6, **kwargs):
    xa = x < a
    xb = x > b
    idx = xa | xb
    d = np.zeros_like(x)
    d[~idx] = 1 / (b - a + 1)
    return d

def EX(x, f, **kwargs):
    def m(x, **kwargs):
        return x
    return general_expectation(x, f, m, **kwargs)

a = -13
b = 6873
SU = np.arange(a, b + 1, dtype = np.float64)
EX(SU, uniform_density, a = a, b = b)

3430.0

def VX(x, f, **kwargs):
    def v(x, **kwargs):
        return (x - EX(x, f, **kwargs)) ** 2
    return general_expectation(x, f, v, **kwargs)

VX(SU, uniform_density, a = a, b = b)

3952564.0

((b - a + 1) ** 2 - 1) / 12

3952564.0

def PX(x, f, **kwargs):
    def px(x, **kwargs):
        return np.isin(x, kwargs["A"]) # x == kwargs["A"]
    return general_expectation(x, f, px, **kwargs)

PX(SU, uniform_density, a = a, b = b, A = np.array([1., 2, 3]))

0.00043560331058516046