Expected Frequency of digit occurrences in leading 2 digits per Benford's Law

benfords_law_expected.py

# Probability of digit (d) occurring in nth position
    def calc_expected_probability(d: int, n: int) -> float:
        # generalization of Benford's Law ~ Summation( log[ 1 + (1/(10k+d)] ) where d = digit, k = (10^(n-2), 10^(n-1))
        # source: https://en.wikipedia.org/wiki/Benford%27s_law
        prob = 0
        if (d == 0) and (n == 1):
            prob = 0
        elif n == 1:
            prob = math.log10(1 + (1 / d))
        else:
            l_bound = 10 ** (n - 2)
            u_bound = 10 ** (n - 1)
            for k in range(l_bound, u_bound):
                prob += math.log10(1 + (1 / (10 * k + d)))
        return round(prob, 3)

    # populate matrix with Benford's Law (10x2 matrix for r= digit, c=location - 1)
    expected_prob_matrix = np.zeros((10, 2))
    for d_i in range(expected_prob_matrix.shape[0]):
        for n_i in range(expected_prob_matrix.shape[1]):
            expected_prob_matrix[d_i, n_i] = calc_expected_probability(d_i, n_i + 1)