Probability (as a limit point)

import numpy as np
import matplotlib.pyplot as plt

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

x = np.array([1., 3, 6, 10]) # still doesn't work, but I know why
uniform_density(x)

array([0.16666667, 0.16666667, 0.16666667, 0.        ])

x1 = x < 1 
x6 = x > 6
x1, x6

(array([False, False, False, False]), array([False, False, False,  True]))

idx = x1 | x6
~idx

array([ True,  True,  True, False])

np.arange(np.size(x))[~idx]

array([0, 1, 2])

array([ 1.,  3.,  6., 10.])

np.ones_like(x)

array([1., 1., 1., 1.])

np.zeros_like(x)

array([0., 0., 0., 0.])

\[\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N 1_A(x_n) = \mathbb{P}[X \in A]\]

rng = np.random.default_rng()
N = 1000
x = rng.integers(1, 7, size = N) # Uniform(1, 6)

np.mean(x == 6)

0.164

ndx = np.arange(1, N + 1)
cm = np.cumsum(x == 6) / ndx # mean of data == 6
plt.plot(ndx, cm);

ndx = np.arange(1, N + 1)
m = (6 + 1) / 2 # mean of distribution
v = ((6 - 1 + 1) ** 2 - 1) / 12 # variance of distribution
cm = np.cumsum((x - m) ** 2) / ndx # variance of data
plt.plot(ndx, cm);
plt.axhline(v, color = "black");

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