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    Tutorial 3

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    Trân Lê
    The document discusses tests for evaluating the randomness of random number generators (RNGs), specifically the Kolmogorov-Smirnov test and chi-square test. It provides details on how to perform each test, including ranking data, calculating test statistics, determining critical values, and deciding whether to reject or fail to reject the null hypothesis that the numbers are uniformly distributed between 0 and 1 based on comparing the test statistic to the critical value. Examples are given demonstrating applications of the Kolmogorov-Smirnov test to various datasets to determine if they can be considered random numbers from a uniform distribution.

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    Tutorial 3

    Uploaded by

    Trân Lê
    56 views4 pages
    The document discusses tests for evaluating the randomness of random number generators (RNGs), specifically the Kolmogorov-Smirnov test and chi-square test. It provides details on how to perform each test, including ranking data, calculating test statistics, determining critical values, and deciding whether to reject or fail to reject the null hypothesis that the numbers are uniformly distributed between 0 and 1 based on comparing the test statistic to the critical value. Examples are given demonstrating applications of the Kolmogorov-Smirnov test to various datasets to determine if they can be considered random numbers from a uniform distribution.

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    The document discusses tests for evaluating the randomness of random number generators (RNGs), specifically the Kolmogorov-Smirnov test and chi-square test. It provides details on how to perform each test, including ranking data, calculating test statistics, determining critical values, and deciding whether to reject or fail to reject the null hypothesis that the numbers are uniformly distributed between 0 and 1 based on comparing the test statistic to the critical value. Examples are given demonstrating applications of the Kolmogorov-Smirnov test to various datasets to determine if they can be considered random numbers from a uniform distribution.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    56 views4 pages

    Tutorial 3

    Uploaded by

    Trân Lê
    The document discusses tests for evaluating the randomness of random number generators (RNGs), specifically the Kolmogorov-Smirnov test and chi-square test. It provides details on how to perform each test, including ranking data, calculating test statistics, determining critical values, and deciding whether to reject or fail to reject the null hypothesis that the numbers are uniformly distributed between 0 and 1 based on comparing the test statistic to the critical value. Examples are given demonstrating applications of the Kolmogorov-Smirnov test to various datasets to determine if they can be considered random numbers from a uniform distribution.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 4

    NETW 707: Modeling & Simulation

    Assoc. Prof. Tallal El-Shabrawy


    Tutorial 3

    Tests for Random Number Generators


    1. A random number generator gave the following numbers
    U= {0.615, 0.79, 0.92, 0.74, 0.18}
    Use the Kolmogorov Smirnov- test with a level of significance  of 0.01 to determine if the
    hypothesis that the number in U are uniformly distributed on [0, 1) can be rejected or not.
    2. Suppose that five numbers: 0.44, 0.81, 0.14, 0.05 and 0.93 were generated and it is desired to
    perform a test on uniformity using the Kolmogorov Smirnov- test with a level of significance
     of 0.05.
    3. Can the following five numbers be regarded as a random choice from the interval 0 to 1: 0.52,
    0.65, 0.13, 0.71, and 0.58? Use Kolmogorov Smirnov- test with a level of significance  of
    0.05.
    4. A sample of size 10 is obtained with: 0.503, 0.621, 0.447, 0.203, 0.701, 0.480, 0.320, 0.581,
    0.55, and 0.381 for a level of significance   5 % . Test the null hypothesis that the numbers
    in a sample are uniformly distributed on [0, 1) can be rejected or not, using Kolmogorov
    Smirnov- test.
    5. The sequence of numbers: 0.54, 0.73, 0.98, 0.11, and 0.68 has been generated. Use the
    Kolmogorov Smirnov- test with a level of significance   0.05 to determine if the hypothesis
    that the numbers are uniformly distributed on the interval [0, 1) can be rejected or not.
    6. Use the Chi-square test with α=0.05 to test whether the data shown next is uniformly
    distributed
    NETW 707: Modeling & Simulation
    Assoc. Prof. Tallal El-Shabrawy
    Tutorial 3
    Testing Random – Number Generators

    Since RNGs are completely deterministic, we need to test them to see if they appear to be
    random and IID uniform on [0, 1]. Most computers will have built-in RNGs but unless software
    is designed for simulation the RNGs can be extremely poor and may not be adequate to be used to
    generate random variates in a stochastic model. It is recommended to test such generators before
    usage (or any generator in general) with one of the following tests. There two types of tests:
     empirical tests
     theoretical tests
    In the empirical test the Ui’s produced by the generator themselves are used to check how
    good is the generator using any of the statistical tests available.
    The theoretical test on the other hand uses the parameters of the generator to assess it
    globally without actually generating any random numbers. So this test is not actually a test in the
    statistical sense.

    The Kolmogrov-smirnov test


    This test compares the continuous CDF, F(x) of the uniform distribution to the empirical
    distribution SN(x) from the data sample.
    - BY definition the uniform distribution have CDF F(x) as follows : F  x   x, 0  x 1
    - If the sample from the random generator is [R1, R2, …, RN] then the empirical CDF SN(x) is
    Number of Rs  x
    given by : S N x  
    N
    - The test is simply based on the largest absolute deviation between F(x) and SN(x) over the
    range of the random variable
    - Quantitatively this means it is based on : D  max | F ( x)  S N ( x) |
    - Now D in itself is a random variable with a known distribution as a function of N and given as
    follows
    For testing we follow the following simple steps :
    1. Rank the data from smallest to largest.

    2. Compute D+, D- given by : D  max  i
    1i  N N

     R(i) ,
    1i  N

    D   max R(i)  iN1 
    3. Compute D=max (D+, D- )
    i. Determine the critical value D from the table. For significant level  and sample size N.
    ii. If D  D . Then no difference has been detected between the sample distribution and the
    uniform distribution.
    The Chi-square Test

    1- Rank the data from smallest to largest


    R1≤R2≤…..≤RN

    2- Divide the range RN- R1 in n equidistant intervals such that each interval has at least 5
    observations

    3- Calculate
    where:
    Oi = the number of observations in the ith class
    Ei = the expected number in the ith class
    n = the number of classes

    4- For significant level α, utilize the table of percentage points of the Chi square distribution
    with v degrees of freedom to determine

    5- If ≤
    Accept: “No difference between and F(x)
    If >
    Reject

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