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Tribhuvan university CSIT 2065 Syllabus Simulation and modeling Old Questions 2069

Simulation and modeling - Old Questions

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3.  What are the properties of random number? The sequence of numbers 0.54, 0.73, 0.98, 0.11 and 0.68 has been generated. Use the Kolmogorov-Smirnov test α=0.05 to determine if the hypothesis that the numbers are uniformly distributed on the interval 0 to 1 can be rejected. (Note that the critical value of D for α=0.05 and N=5 is 0.565).

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Asked in 2069

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Random numbers are samples drawn from a uniformly distributed random variable between some satisfied intervals, they have equal probability of occurrence.

Properties of Random Numbers

1. Uniformity:

  • The random numbers generated should be uniform. That means a sequence of random numbers should be equally probable every where.
  • If we divide all the set of random numbers into several numbers of class interval then number of samples in each class should be same.
  • If ‘N’ number of random numbers are divided into ‘K’ class interval, then expected number of samples in each class should be equal to ei = N / K.

2. Independent:

  • Each random number should be independent samples drawn from a continuous uniform distribution between 0 and 1.
  • The probability density function is given by:
    f(x) = 1, 0 <= x <= 1
          = 0, otherwise

3. Maximum Density:

  • The large samples of random number should be generated in a given range.

4. Maximum Cycle:

  •  It states that the repetition of numbers should be allowed only after a large interval of time.

Now,

Given sequence of number,

    0.54, 0.73, 0.98, 0.11 and 0.68

Arranging the given number in ascending order:

    0.11, 0.54, 0.68, 0.73, 0.98

Here, N = 5

Calculation table for Kolmogorov-Smirnov test :

i



10.110.20.090.11
20.540.4-0.34
30.680.6-0.28
40.730.80.070.13
50.9810.020.18

Now, calculating

\\begin{displaymath}D^+ = {\\rm max}_{1 \\le i \\le N} \\left\\{ \\frac{i}{N} - R_{(i)} \\right\\} \\end{displaymath} = 0.09

\\begin{displaymath}D^- = {\\rm max}_{1 \\le i \\le N} \\left\\{ R_{(i)} - \\frac{i-1}{N} \\right\\} \\end{displaymath} = 0.35

$D = {\\rm max} (D^+, D^-)$ = 0.35

Given, Critical value $D_\\alpha$ = 0.565

Since the computed value, D = 0.35, is less than the tabulated critical value, $D_\\alpha$ = 0.565, the hypothesis of no difference between the distribution of the generated numbers and the uniform distribution is not rejected.