Bsc CSIT Semester 5 – Simulation and Modeling – Unit 5. Random Numbers

Define true random numbers and pseudo random numbers with its properties. The sequence of numbers 0.64, 0.25, 0.58, 0.72, 0.90 has been generated. Use KS Test with Dα=0.050 =>0.512 to determine if the hypothesis that they are uniformly distributed on interval [0,1] can be rejected.

Use multiplicative congruential method to generate a sequence of random numbers with x₀ = 7, a = 11, m = 16.

Define and develop a Poker test for four-digit random numbers. A sequence of 1,000 random numbers, each of four digits has been generated. The analysis of the numbers reveals that in 525 numbers all four digits are different, 419 contain exactly one pair of like digits, 47 contain two pairs, 1 have three digits of a kind and 7 contain all like digits. Use Poker test to determine whether these numbers are independent. (Critical value of chi-square for d.f. = 0.05 and N = 4 is 9.49).

Generate ten 3 digit random integers and corresponding random variables using Multiplicative Congruential method where a = 7, and X₀ = 22.

What are the two main properties of random numbers? Test whether the 3rd, 7th, 11th, and so on numbers in the sequence in the following random number sample are auto-correlated. (Z₀ = 0.05 and Z₀.₀₂₅ = 1.96)0.12 0.01 0.23 0.28 0.89 0.31 0.64 0.28 0.83 0.93 0.99 0.15 0.33 0.35 0.91 0.41 0.60 0.27 0.75 0.88 0.68 0.49 0.05 0.43 0.95 0.58 0.19 0.36 0.69 0.87

Generate 10 random integers using Linear congruential method where m = 1000, a = 19, c = 6, and X₀ = 13.

Explain the Monte Carlo simulation method with example.

Write short notes:

a) Non-stationary Poisson Process

b) Poker Test

List the types of random number. Discuss about uniformity testing.

Describe about the use of Monte Carlo with an example.

Explain the random variate generation using inverse transform technique.

What are the properties of random number? The sequence of numbers 0.23, 0.45, 0.67, 0.12, 0.89, 0.34, 0.56, 0.78, 0.19, 0.41, 0.63, 0.08, 0.85, 0.29, 0.51, 0.73, 0.16, 0.94, 0.37, 0.59 has been generated. Test for the independence among numbers in the sequence starting with index (i) = 2 and lag (m)=3 using auto-correlation test. (α = 0.05, Zα/2=1.96)

Discuss the inverse transform technique and acceptance-rejection technique for random variate generation.