Benford distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ Benfordis used to indicate that the random variable X has the Benford distribution. A Benford random variable X has probability mass function

f(x) = log_{10}( 1 + \frac{1}{x}) x = 1,2,...,9.

The probability mass function is illustrated below.

PSfrag replacements

x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = log_{10}(1 + x) x = 1,2,...,9.

The survivor function on the support of X is

S(x) = P(X ≥ x) = 1 − log_{10}x x = 1,2,...,9.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{log_{10}(1 + \frac{1}{x})}{1 − log_{10}x} x = 1,2,...,9.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = −ln(1 − log_{10}x) x = 1,2,...,9.

The inverse distribution function is

F^{−1} (u) = ⌊10^{u}⌋ 0 < u < 1.

The population mean, variance, skewness, and kurtosis of X is approximated with results from APPL

E[X] = \frac{2ln(2) − 4ln(3) + 8ln(5) − ln(7)}{ln(2) + ln(5)} ≅ 3.4402

V [X] ≅ 6.0565  E [( \frac{X − μ}{σ} ) ^{3}] ≅ 0.7956  E [( \frac{X − μ}{σ} ) ^{4}] ≅ 2.4518.

APPL verification: The APPL statements

X := BenfordRV(); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

verify the population mean, variance, skewness, kurtosis and moment generating function.