Principles for Generating Non-Uniform Random Numbers

In many simulations random numbers from a given distribution function F(x) with density f(x) = F′(x) are necessary. If f(x) = 1 for 0 ≤ x ≤ 1 uniformly distributed random numbers are needed. For this purpose the linear congruential method is widely used: A sequence of integers is initialized with a value z₀ and continued as $$ {z_{{i + 1}}} \equiv a{z_i} + r\;(\bmod \,m),\;0 \leqslant {z_i} < m\;for\,all\;i $$ The fractions u_i = z_i/m are the derived pseudo-random numbers in the interval [0,1). The constants m, the modulus, a, the multiplicator, r, the increment, and z₀, the starting number are suitably chosen non-negative integers. Three choices of m, a and r are common on most computers: 1.r = 0, m = 2E, a ≡ 5 (mod 8) and z0 ≡ 1 (mod 4). All zi ≡ 1 (mod 4) are generated.2.r = 0, m = p, p prime, a a primitive root mod p. All zi = 1,..., p − 1 are generated.3.gcd(r, m) = 1, m = 2E, a = 1 (mod 4). All integers 0, 1,..., 2E − 1 are generated.