--- title: "Properties and generation techniques of RN" subject: "Simulation and Modeling" chapter: 6 key: "semester_8/simulation_and_modeling:rn-properties-generation-techniques" source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/rn-properties-generation-techniques site: BCT Engineering --- # Properties and generation techniques of RN Simulation and Modeling · Chapter 6: Random Number - Asked in: 2074 Bhadra, 2+6 marks - Video: [Random Numbers -System Modelling & Simulation |Properties of Random Numbers |Generation of Pseudo RN](https://www.youtube.com/watch?v=otuwEdCRoww) — Infinite Love - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/rn-properties-generation-techniques - Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-6 ## Question What are the properties of random number? Explain the techniques of generation of random number with appropriate example. ## Short answer Random numbers require uniformity and independence; generators should be fast, reproducible, and have long periods. Techniques include mid-square, mid-product, linear congruential, multiplicative congruential, additive congruential, and physical sources, each illustrated with examples such as LCG a=5, c=3, m=16. ## Answer #### Properties of random numbers 1. **Uniformity**: numbers are uniformly distributed over [0, 1); each of n equal sub-intervals gets about N/n of N numbers. 2. **Independence**: each number is unrelated to the previous ones. A good generator should also be fast, use little memory, have a long period (full cycle), be reproducible from a seed and pass statistical tests. #### Generation techniques Computer-generated (pseudo-random) numbers are produced by a formula. R = X/m maps integers to [0, 1). **1. Mid-square method.** Square the seed, take the middle digits. Seed 1234: 1234^2 = 01522756, so next is 5227 (R = 0.5227); 5227^2 = 27321529 gives 3215 (0.3215); then 3362. It can quickly degenerate: seed 2100 gives 4100, 8100, 6100, 2100 and repeats. **2. Mid-product method.** Multiply two consecutive numbers and take the middle digits: 5731 x 7182 = 41160042 gives 1600; 7182 x 1600 = 11491200 gives 4912. **3. Linear congruential method.** X(n+1) = (a X(n) + c) mod m. Example: a = 5, c = 3, m = 16, X0 = 7. | n | X(n) | R = X/16 | |---|---|---| | 1 | 6 | 0.375 | | 2 | 1 | 0.0625 | | 3 | 8 | 0.5 | | 4 | 11 | 0.6875 | | 5 | 10 | 0.625 | | 6 | 5 | 0.3125 | | 7 | 12 | 0.75 | | 8 | 15 | 0.9375 | It repeats after 16 numbers (full period). Full period is obtained when c and m are relatively prime, a - 1 is divisible by every prime factor of m, and by 4 if m is divisible by 4. Usually m = 2^b. **4. Multiplicative congruential method.** X(n+1) = a X(n) mod m (c = 0). Example: a = 13, m = 64, X0 = 1 gives 13, 41, 21, 17, 29, 57, 37, 33. **5. Additive congruential method.** X(n) = (X(n-1) + X(n-k)) mod m. With k = 2, m = 100 and seeds 65, 89: 54, 43, 97, 40, 37. **6. Other:** built-in library functions, random number tables and physical (hardware) generators. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. If [0,1) is divided into n equal classes and N numbers are generated, how many should fall in each class? A. About N/n B. About N x n C. About n/N D. Exactly zero or one Answer: A. Uniformity means each class gets about N/n of the N numbers. (Key point: Uniformity) 2. Which is a condition for full period of a linear congruential generator? A. c and m must be relatively prime B. c must be 0 C. m must be odd D. a must equal m Answer: A. Full period needs c and m relatively prime, a - 1 divisible by every prime factor of m (and by 4 if 4 divides m). (Key point: Full period conditions) 3. For a = 5, c = 3, m = 16, X0 = 7, what are X1 and R1? A. X1 = 38, R1 = 2.375 B. X1 = 6, R1 = 0.375 C. X1 = 7, R1 = 0.4375 D. X1 = 1, R1 = 0.0625 Answer: B. (5 x 7 + 3) mod 16 = 38 mod 16 = 6, and R = 6/16 = 0.375. (Key point: LCG numeric example) 4. What distinguishes the multiplicative congruential method from the mixed one? A. It uses c = 0 B. It uses squaring of the seed C. It uses two earlier values D. It uses a table Answer: A. The multiplicative form is X(n+1) = a X(n) mod m, i.e. with increment c = 0. 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