We begin with a historically prominent method.<\/p>\n
Linear Congruential Generator. <\/strong>To generate a sequence of random numbers, start with \\(B_0\\), a starting value that is known as a “seed.” Update it using the recursive relationship For illustrative values of \\(a\\) and \\(m\\), Microsoft’s Visual Basic uses \\(m=2^{24}\\), \\(a=1,140,671,485\\), and \\(c = 12,820,163\\) (see http:\/\/support.microsoft.com\/kb\/231847). This is the engine underlying the random number generation in Microsoft’s Excel program.<\/p>\n The sequence used by the analyst is defined as \\(U_n=B_n\/m.\\) The analyst may interpret the sequence {\\(U_{i}\\)} to be (approximately) identically and independently uniformly distributed on the interval (0,1). To illustrate the algorithm, consider the following.<\/p>\n Example. Take \\(m=15\\), \\(a=3\\), \\(c=2\\) and \\(B_0=1\\). Then we have:<\/p>\n Sometimes computer generated random results are known as pseudo<\/em>-random numbers to reflect the fact that they are machine generated and can be replicated. That is, despite the fact that {\\(U_i\\)} appears to be i.i.d, it can be reproduced by using the same seed number (and the same algorithm). The ability to replicate results can be a tremendous tool as you use simulation while trying to uncover patterns in a business process.<\/p>\n The linear congruential generator is just one method of producing pseudo-random outcomes. It is easy to understand and is (still) widely used. The linear congruential generator does have limitations, including the fact that it is possible to detect long-run patterns over time in the sequences generated (recall that we can interpret “independence” to mean a total lack of functional patterns). Not surprisingly, advanced techniques have been developed that address some of this method’s drawbacks.<\/p>\n
\n\\[
\nB_{n+1} = a B_n + c \\text{ modulo }m, ~~ n=0, 1, 2, \\ldots .
\n\\]
\nThis algorithm is called a linear congruential generator<\/em>. The case of \\(c=0\\) is called a multiplicative<\/em> congruential generator; it is particularly useful for really fast computations.<\/p>\n\n
\n step \\(n\\)<\/th>\n \\(B_n\\)<\/th>\n \\(U_n\\)<\/th>\n<\/tr>\n \n 0<\/td>\n \\(B_0=1\\)<\/td>\n <\/td>\n<\/tr>\n \n 1<\/td>\n \\(B_1 =\\mod(3 \\times 1 +2) = 5\\)<\/td>\n \\(U_1 = \\frac{5}{15}\\)<\/td>\n<\/tr>\n \n 2<\/td>\n \\(B_2 =\\mod(3 \\times 5 +2) = 2\\)<\/td>\n \\(U_2 = \\frac{2}{15}\\)<\/td>\n<\/tr>\n \n 3<\/td>\n \\(B_3 =\\mod(3 \\times 2 +2) = 8\\)<\/td>\n \\(U_3 = \\frac{8}{15}\\)<\/td>\n<\/tr>\n \n 4<\/td>\n \\(B_4 =\\mod(3 \\times 8 +2) = 11\\)<\/td>\n \\(U_4 = \\frac{11}{15}\\)<\/td>\n<\/tr>\n<\/table>\n