--- title: "Pseudo-random; Kolmogorov-Smirnov numerical" subject: "Simulation and Modeling" chapter: 6 key: "semester_8/simulation_and_modeling:ks-test-uniform-numerical" source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/ks-test-uniform-numerical site: BCT Engineering --- # Pseudo-random; Kolmogorov-Smirnov numerical Simulation and Modeling · Chapter 6: Random Number - Asked in: 2071 Bhadra, 2+6 marks - Video: [Simulation Modeling 30 The Kolmogorov-Smirnov Test](https://www.youtube.com/watch?v=NzE5anmP6JE) — İsmail Başoğlu - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/ks-test-uniform-numerical - Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-6 ## Question Define pseudo random numbers. The following numbers have been generated 0.44, 0.19, 0.88, 0.27, 0.55, 0.13, 0.63, 0.74, 0.11 and 0.33. Use the Kolmogorov-Smirnov test with α = 0.05 to determine, if the hypothesis that the numbers are uniformly distributed on the interval [0, 1] can be rejected. (Note that the critical value of D for α 0.05 and N = 10 is 0.410.) ## Short answer Pseudo-random numbers are numbers generated by a deterministic algorithm that mimic true random numbers, reproducible from a seed, uniformly distributed on [0,1) and passing statistical tests. For the K-S test, D = 0.17, less than critical value 0.410 (α=0.05, N=10), so the uniformity hypothesis cannot be rejected. ## Answer #### Pseudo-random numbers Pseudo-random numbers are numbers generated by a deterministic algorithm (such as the linear congruential method Xn+1 = (aXn + c) mod m) that imitate true random numbers. They are designed to be uniformly distributed on [0, 1) and to pass statistical tests of uniformity and independence, but the sequence can be reproduced from the seed and eventually cycles. #### K-S test H0: the numbers are uniformly distributed on [0, 1]. H1: they are not. N = 10, α = 0.05. Rank the numbers and compute the differences: | i | R(i) | i/N | i/N − R(i) | R(i) − (i−1)/N | |---|---|---|---|---| | 1 | 0.11 | 0.1 | -0.01 | 0.11 | | 2 | 0.13 | 0.2 | 0.07 | 0.03 | | 3 | 0.19 | 0.3 | 0.11 | -0.01 | | 4 | 0.27 | 0.4 | 0.13 | -0.03 | | 5 | 0.33 | 0.5 | 0.17 | -0.07 | | 6 | 0.44 | 0.6 | 0.16 | -0.06 | | 7 | 0.55 | 0.7 | 0.15 | -0.05 | | 8 | 0.63 | 0.8 | 0.17 | -0.07 | | 9 | 0.74 | 0.9 | 0.16 | -0.06 | | 10 | 0.88 | 1.0 | 0.12 | -0.02 | D⁺ = max(i/N − R(i)) = **0.17** (at i = 5 and i = 8) D⁻ = max(R(i) − (i−1)/N) = **0.11** (at i = 1) D = max(D⁺, D⁻) = **0.17** Critical value D0.05 = 0.410 (N = 10). **Conclusion.** Since D = 0.17 < 0.410, the hypothesis of uniformity cannot be rejected. The numbers can be treated as uniformly distributed on [0, 1]. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Pseudo-random numbers are: A. Generated by a deterministic algorithm and repeatable from a seed B. Generated from physical noise C. Not repeatable D. Always independent Answer: A. They come from a deterministic algorithm such as an LCG and are repeatable and eventually cycle. (Key point: Pseudo-random numbers) 2. What must be done first in the K-S calculation? A. Group numbers into classes B. Rank the numbers from smallest to largest C. Square each number D. Compute the mean Answer: B. The numbers are ranked R(1) <= ... <= R(N) before computing D+ and D-. (Key point: First step of K-S) 3. For the given 10 numbers, D+ = 0.17 and D- = 0.11. What is D? A. 0.11 B. 0.28 C. 0.17 D. 0.06 Answer: C. D = max(D+, D-) = 0.17. (Key point: Value of D) 4. With critical value 0.410, what is the conclusion? A. Reject uniformity B. Uniformity cannot be rejected since 0.17 < 0.410 C. Reject because D is positive D. Need more numbers to decide anything Answer: B. D < critical value means no evidence against uniformity. 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