Generation of Random Numbers - Coursework Example

Name: Topic: Tutor : Date: 23rd November, 2015 Generation of random numbers Introduction There are many methods of random number generation especially integers which are in a group but do not have a relationship to each other. The methods that generate random numbers includes; linear congruencies generators, inverse transform method, and many others. After the numbers have been generated there is need of testing the reliability of the number generated in terms of uniformity and impedance. Various methods are used in testing weather the generated random numbers are universally distributed and this include gap test, frequency test, run test, poker test and autocorrelation test. The procedure is somehow complicated. It involves finding the expected frequencies for each class of frequency distribution by using the standard normal distribution (Gentle 29). The actual frequencies are compared to the expected frequencies, using the chi-squire goodness-of-fit test. If the observed frequencies are close in value to the expected frequencies, chi-squire test value will be small, and the null hypothesis cannot be rejected. In this case, it concluded that the variable is approximately normally distribution. In this case, it can be concluded that the variable is not normally distributed(Gentle 42). In this paper inverse method of random number generation will be discussed and how it was used to generate uniform distributed numbers. Then 500 numbers will be generated and later, tests for randomness will be done. Method of generating random numbers The method selected generating uniformly distributed random numbers between 0 and I Monte Carlo method which excels function of RAND will be used. In Monte Carlo method one evaluates a non-random quantity as an expectation of a random variable. In order to apply the technique it is therefore necessary to use a large number of random numbers from a specified distribution. This is incorporated into excel approximations. The Monte Carlo method requires large random numbers are often required and problems of periodicity in a finite-length table can therefore arise. For this reason one needs a better source of random numbers and it is typical to use pseudo-random numbers that are generated by random number generating algorithms (RNGs). RNGs have the great advantage of being easily incorporated into the coding of Monte Carlo simulations, producing an unlimited supply of random variates quickly and without resorting to physical means. They have the further advantage of being reproducible: if the same seed is given at the beginning of two runs of the RNG, identical sequences of random variates will be produced. Exact reproductions of individual simulations may be very usefully when debugging the code and the advantage of having to store only a single seed rather than a very large sequence of random numbers is clear. In what follows we shall refer to realizations of a random variable generated by a computer as random variates to distinguish them from truly random numbers. This helps in generation of random variates from the uniform distribution. In what follows it is assumed that a source of random variates from the -distribution has been established and the question of how to use these to generate random variates from other distributions is considered. The Monte Carlo method provides probabilistic algorithms for simulating systems where underlying randomness exists (Kunh and Johnsons,136). Monte Carlo simulation is usually undertaken to determine either the expected value or the distribution of a random variable   connected with a particular stochastic model. A simulation of the model results in the output data  , a realisation of the random variable . A second simulation (with an independent set of random variates) provides a new output. The simulations continue until we have accumulated a sufficiently large number  of outputs  . The arithmetic average of these outputs   is then used as an estimator of . Any other features of the distribution of  may also be estimated using the corresponding features of the simulated data. The data analysis add-in excel has a feature to generate random numbers from a specific probability distribution. In this case a list of 500 random numbers from uniform distribution. This is done as follows; 1. Open a new worksheet and select tool>data analysis>random number generation from analysis tools. Click ok. 2. In the dialog box. Type 1 for the number of variables. Leave the number of random numbers black. 3. For distribution select uniform. 4. In the parameter box. type 0 for the lower bound and 1 for the upper bound. 5. You may type in an integer value between 0 and 1 for the random seed. For this example, type 1 for random seed. 6. Select output range and type in A1:A500. Click ok. To cover the random numbers to a list of the integers: 7. Select cell B1 and select the past function icon from the toolbar. 8. Select the math & trig function category and scroll to the function name INT to cover the data in column A to integers. Note that the INT function rounds the argument (input) down to the nearest integer. 9. Type cell A 1 for the number in the INF dialog box. Click ok. 10. While cell B1 is selected in the worksheet, move the point to the lower right –hand corner of the cell until a thick plus sign appears. Right click on the mouse and drag the plug down to cell B500; then release the mouse key. 11. The number from column A should have been rounded to integers in column B. Here is a sample of the data produced from the preceding procedures. Generation of 500 random numbers The numbers were generated using excel function of RAND. The numbers generated are as follows; x=rand(42,12) x = 0.63 0.45 0.55 0.10 0.76 0.57 0.98 0.87 0.50 0.97 0.36 0.00 0.36 0.23 0.58 0.04 0.41 0.68 0.72 0.93 0.69 0.57 0.45 0.77 1.00 0.80 0.51 0.56 0.49 0.36 0.84 0.67 0.83 1.00 0.39 0.85 0.22 1.00 0.08 0.77 0.69 0.62 0.43 0.21 0.61 0.55 0.78 0.92 0.65 0.03 0.72 0.31 0.97 0.81 0.47 0.65 0.57 0.52 0.73 1.00 0.61 0.54 1.00 0.18 0.33 0.02 0.56 0.07 0.33 0.33 0.43 0.51 0.39 0.09 0.35 0.34 0.84 0.08 0.27 0.41 0.46 0.43 0.69 0.27 0.14 0.80 0.97 0.21 0.74 0.97 0.75 0.67 0.71 0.49 0.95 0.10 0.03 0.99 0.35 0.51 0.95 0.65 0.50 0.93 0.88 0.07 0.78 0.51 0.42 0.07 0.89 0.91 0.03 0.23 0.65 0.81 0.72 0.89 0.71 0.59 0.18 0.94 0.45 0.63 0.34 0.40 0.31 0.48 0.02 0.06 0.11 0.76 0.73 0.02 0.41 0.10 0.66 0.12 0.14 0.76 0.67 0.44 0.39 0.08 0.37 0.68 0.22 0.39 0.28 0.27 0.48 0.42 0.44 0.83 0.59 0.66 0.84 0.78 0.13 0.05 0.23 0.26 0.36 0.97 0.44 0.39 0.46 0.52 0.73 0.53 0.31 0.50 0.71 0.33 0.79 0.99 0.12 0.61 0.05 0.17 0.57 0.89 0.73 0.43 0.62 0.15 0.78 0.86 0.81 0.82 0.23 0.94 0.18 0.90 0.78 1.00 0.59 0.35 0.67 0.39 0.32 0.89 0.83 0.59 0.99 0.63 0.70 0.81 0.66 0.12 0.13 0.45 0.25 0.93 0.02 0.44 0.27 0.14 0.01 0.49 0.05 0.88 0.02 0.25 0.34 0.19 0.86 0.94 0.92 0.22 0.84 0.89 0.35 0.09 0.57 0.78 0.38 0.26 0.08 0.64 0.22 0.18 0.92 0.14 0.45 0.93 0.30 0.88 0.55 0.90 0.67 0.45 0.37 0.04 0.77 0.39 0.24 0.40 0.94 0.91 0.56 0.59 0.50 0.84 0.09 0.11 0.04 0.93 0.72 0.05 0.98 0.56 0.40 0.50 0.22 0.53 0.64 0.62 0.38 0.92 0.86 0.34 0.29 0.60 0.40 0.61 0.57 0.55 0.18 0.94 0.70 0.71 0.28 0.74 0.80 0.15 0.52 0.82 0.12 0.68 0.05 0.35 0.73 0.62 0.73 0.80 0.90 0.90 0.66 0.53 0.67 0.37 0.72 0.41 0.22 0.34 0.14 0.54 0.60 0.45 0.95 0.20 0.60 0.24 0.35 0.98 0.27 0.95 0.84 0.69 0.88 0.21 0.72 0.45 0.06 0.58 0.66 0.95 0.67 0.12 0.14 0.89 0.94 0.90 0.40 0.43 0.06 0.87 0.38 0.68 0.48 0.73 0.59 0.05 0.55 0.76 0.83 0.97 0.15 0.41 0.63 0.99 0.62 0.65 0.37 0.30 0.73 0.88 0.13 0.62 0.02 0.11 0.02 0.77 0.24 0.83 0.81 0.05 0.58 0.29 0.06 0.70 0.43 0.44 0.91 0.34 0.18 0.40 0.50 0.20 0.03 0.67 0.08 0.72 0.83 0.30 0.80 0.66 0.83 0.75 0.49 0.72 0.45 0.66 0.16 0.35 0.61 0.40 0.75 0.24 0.77 0.84 0.88 0.72 0.65 0.12 0.32 0.52 0.52 0.83 0.81 0.30 0.93 0.32 0.35 0.88 0.52 0.41 0.30 0.56 0.86 0.40 0.38 0.68 0.11 0.55 0.45 0.58 0.37 0.28 0.01 0.16 0.10 0.39 0.62 0.53 0.18 0.98 0.96 0.07 0.94 0.72 0.54 0.56 0.91 0.36 0.58 0.41 0.10 0.55 0.04 0.92 0.83 0.28 0.10 0.69 0.11 0.53 0.60 0.49 0.33 0.97 0.80 0.85 0.90 0.15 0.43 0.52 0.28 0.75 0.19 0.62 0.19 0.29 0.37 0.83 0.63 0.84 0.14 0.25 0.58 0.90 0.36 0.67 0.54 0.59 0.39 0.86 0.73 0.56 Tests for randomness and independence of the numbers generated The numbers generated needs to be tested for the reliability through independence test and randomness. The tests to do this are run test, poker test, frequency test, Autocorrelation test and Gap test. These tests are described below; The run test- Run test is a test that is carried out to determine the arrangement of the numbers randomly selected in order to certain their independence. It is successfully done by selecting different numbers to carry out the test(Xiannong 1). In our case where 0 and 1 are numbers that are selected, the sequence of numbers will be different such as: 0.30 0.88 0.55 0.90 0.67 0.45 0.73 0.80 0.90 0.90 0.66 0.53 0.67 0.37 0.97 0.80 0.85 0.90 0.15 0.43 0.30 0.88 0.55 0.90 0.67 0.45 0.73 0.80 0.90 0.90 0.66 0.53 0.67 0.37 0.97 0.80 0.85 0.900.150.43 Run test considers the sequence of the selected random number however successful runs depends on the length of run and the number of runs done. Few runs may not give proper result. This method considers the hypothesis of the runs by considering the mean or standard deviation of the runs. The mean and standard deviation of run is calculated using the two formulas shown below, + In testing the hypothesis z value of standardized normal distribution is used as follows: That is Zo=(Xiannong 1) In this case rejections of the independence of the selected data take place when the calculated z value is not within the limits of the selected level of significance. Frequency test- This test is important for selected random numbers as it consider uniformity of the sampled numbers. This test is usually done using chi-squire test and Kolmogorov-Smirnov test which uses null hypothesis. Kolmogorov-Smirnov test looks at the differences in uniformity of the sample observation and the general population (Gentle 48). The difference in uniform distribution of population and sample is usually calculated using the following formulas D=max (Xiannong 1) This is where F(x)=x is for the general population and SN (x) is for the sample. This is called Kolmogorov-Smirnov test. The above variables are calculated using the following; F(x)=x 0 SN (x)= (Xiannong 1) When N > then SN (x) is close to F(x). In testing the hypothesis if D becomes greater than the critical value a conclusion is made that the data of the sample is not uniformly distributed as in the case of the population. Gap Test – this measures interval between the same number and its recurrence. It shows the distance between the same digits (Gentle 26). The gap or real numbers is the set of all real numbers between two numbers, greater than some one number or less than some one number. More exactly, if a and b are real numbers, [a, b] is the set of real numbers x such that ahe gap [a, b] is close since it includes its end point a. Thus (a, b) is the open gap which is the set of all numbers x such that a as would be expected (a, b) as half-open, half-closed gap, etc. (- is the set of all the real numbers x such that x 1.96 There is enough evidence to reject the hypothesis that the numbers were selected at random. Frequency test- In this case we are using Kolmogorov-Smirnov test where we begin to rank the numbers from the smallest to largest and in our case it is numbers 0.30 0.88 0.55 0.90 0.67 0.45 0.73 0.80 0.90 0.90 0.66 0.97 0.80 0.85 0.90 0.15 0.43 0.53 0.67 0.37 , , Then we calculate , = -2.05 Then we go ahead to compute D=max ( D=max ( Then we go ahead to compute D=max ( D=max ( From the critical value for sample size at significance level of 5% D is 14 and 28 Since calculated D (significance D) we conclude that the data from a uniform distribution and conclude that there is no evidence to reject the null hypothesis that the random numbers were from a uniform distribution. Gap test - Since we looking at the gap of two numbers randomly selected 500 times then Let us consider Gap Length Frequency Relative Frequency Cum. Relative Frequency F(x) |F(x) - SN(x)| 0.88 1 1 0.07 0.0696 0.0004 0.55 1 2 0.14 0.1396 0.0004 0.90 4 6 0.43 0.4258 0.0042 0.67 1 7 0.50 0.496 0.004 0.45 1 8 0.57 0.5678 0.0022 0.73 1 9 0.64 0.6396 0.0004 0.80 2 11 0.79 0.7996 0.0004 0.66 1 12 0.86 0.859 0.001 0.85 1 13 0.93 0.929 0.001 0.97 1 14 1.00 0.999 0.001 0.015 The critical value of D is given by D0.05 =1.36/ = 0.061 The calculated D = max |F(x) - SN(x)| is 0.015 which is smaller than D0.05, thus we accept the hypothesis of independence. Poker test – this groups numbers in the series in which they are repeated for example 0.30 0.88 0.55 0.90 0.67 0.45 0.73 0.80 0.90 0.90 0.66 0.97 0.80 0.85 0.90 0.15 0.43 0.53 0.67 0.37 . Therefore, it tests for independence for generated numbers from its frequency. In this case there is 500 possibilities since we have two digits. Comment and conclusion From the findings it will be noted that the generated numbers are from uniformly distributed population as shown by frequency test. Run test shows that the numbers are not biased and they come from the population. Gap test shows that the random number is independence as there value is smaller than the critical value. This leads to a conclusion that data come a uniformly distributed data and there is no interference by human. Works Cited Biebighauser, Dan. Testing Random Number Generators. University of Minnesota - Twin Cities. Jan 2000. Web. 23 November 2015 Gentle, James. Random Number Generation and Monte Carlo Methods (Statistics and Computing). New York: Springer, 2004. Print Gentle, James. Computational Statistics New York: Springer, 2009. Print Givens, Goefs & Hoeting Jenniffer. Computational Statistics. New York: Wiley,2013 Higgins, James. Introduction to Modern Nonparametric Statistics. Massachusetts: Duxbury Press, 2003. Print Kunh, Max & Johnsons, Kjell. Applied Predictive Modeling. New York: Springer, 2013. Print Pareek, Vikas, Ankur Rathi & Divyanjali Sharma. Pseudo Random Number Generation. Saarbrücken: LAP LAMBERT Academic Publishing,2013. Print Xiannong , Meng. Tests for Random Numbers.www. eg.bucknell.edu, June 2002. Web. 23 November 2015. Read More

RNGs have the great advantage of being easily incorporated into the coding of Monte Carlo simulations, producing an unlimited supply of random variates quickly and without resorting to physical means. They have the further advantage of being reproducible: if the same seed is given at the beginning of two runs of the RNG, identical sequences of random variates will be produced. Exact reproductions of individual simulations may be very usefully when debugging the code and the advantage of having to store only a single seed rather than a very large sequence of random numbers is clear.

In what follows we shall refer to realizations of a random variable generated by a computer as random variates to distinguish them from truly random numbers. This helps in generation of random variates from the uniform distribution. In what follows it is assumed that a source of random variates from the -distribution has been established and the question of how to use these to generate random variates from other distributions is considered. The Monte Carlo method provides probabilistic algorithms for simulating systems where underlying randomness exists (Kunh and Johnsons,136).

Monte Carlo simulation is usually undertaken to determine either the expected value or the distribution of a random variable   connected with a particular stochastic model. A simulation of the model results in the output data  , a realisation of the random variable . A second simulation (with an independent set of random variates) provides a new output. The simulations continue until we have accumulated a sufficiently large number  of outputs  . The arithmetic average of these outputs   is then used as an estimator of .

Any other features of the distribution of  may also be estimated using the corresponding features of the simulated data. The data analysis add-in excel has a feature to generate random numbers from a specific probability distribution. In this case a list of 500 random numbers from uniform distribution. This is done as follows; 1. Open a new worksheet and select tool>data analysis>random number generation from analysis tools. Click ok. 2. In the dialog box. Type 1 for the number of variables.

Leave the number of random numbers black. 3. For distribution select uniform. 4. In the parameter box. type 0 for the lower bound and 1 for the upper bound. 5. You may type in an integer value between 0 and 1 for the random seed. For this example, type 1 for random seed. 6. Select output range and type in A1:A500. Click ok. To cover the random numbers to a list of the integers: 7. Select cell B1 and select the past function icon from the toolbar. 8. Select the math & trig function category and scroll to the function name INT to cover the data in column A to integers.

Note that the INT function rounds the argument (input) down to the nearest integer. 9. Type cell A 1 for the number in the INF dialog box. Click ok. 10. While cell B1 is selected in the worksheet, move the point to the lower right –hand corner of the cell until a thick plus sign appears. Right click on the mouse and drag the plug down to cell B500; then release the mouse key. 11. The number from column A should have been rounded to integers in column B. Here is a sample of the data produced from the preceding procedures.

Generation of 500 random numbers The numbers were generated using excel function of RAND. The numbers generated are as follows; x=rand(42,12) x = 0.63 0.45 0.55 0.10 0.76 0.57 0.98 0.87 0.50 0.97 0.36 0.00 0.36 0.23 0.58 0.04 0.41 0.68 0.72 0.93 0.69 0.57 0.45 0.77 1.00 0.80 0.51 0.56 0.49 0.36 0.84 0.67 0.83 1.00 0.39 0.85 0.22 1.00 0.08 0.77 0.69 0.62 0.43 0.21 0.61 0.55 0.78 0.92 0.65 0.03 0.72 0.31 0.97 0.81 0.47 0.65 0.57 0.52 0.73 1.00 0.61 0.54 1.00 0.18 0.33 0.02 0.56 0.07 0.33 0.33 0.43 0.51 0.39 0.09 0.35 0.34 0.84 0.08 0.27 0.41 0.46 0.43 0.69 0.27 0.14 0.80 0.97 0.21 0.74 0.97 0.75 0.67 0.71 0.49 0.95 0.10 0.03 0.99 0.35 0.51 0.95 0.65 0.50 0.93 0.88 0.07 0.78 0.51 0.42 0.07 0.89 0.91 0.03 0.23 0.65 0.81 0.72 0.89 0.71 0.59 0.18 0.94 0.45 0.63 0.34 0.40 0.31 0.48 0.02 0.06 0.11 0.76 0.73 0.02 0.41 0.10 0.66 0.12 0.14 0.76 0.67 0.44 0.39 0.08 0.37 0.68 0.22 0.39 0.28 0.27 0.48 0.42 0.44 0.83 0.59 0.66 0.84 0.78 0.13 0.05 0.23 0.26 0.36 0.97 0.44 0.39 0.46 0.52 0.73 0.53 0.31 0.50 0.71 0.33 0.79 0.99 0.12 0.61 0.05 0.17 0.57 0.89 0.73 0.43 0.62 0.15 0.78 0.86 0.81 0.82 0.23 0.94 0.18 0.90 0.78 1.00 0.59 0.35 0.67 0.39 0.32 0.89 0.83 0.59 0.99 0.63 0.70 0.81 0.66 0.12 0.13 0.45 0.25 0.93 0.02 0.44 0.27 0.14 0.01 0.49 0.05 0.88 0.02 0.25 0.34 0.19 0.86 0.94 0.92 0.22 0.84 0.89 0.35 0.09 0.57 0.78 0.38 0.26 0.08 0.64 0.22 0.18 0.92 0.14 0.45 0.93 0.30 0.88 0.55 0.90 0.67 0.45 0.37 0.04 0.77 0.39 0.24 0.40 0.94 0.91 0.56 0.59 0.50 0.84 0.09 0.11 0.04 0.93 0.72 0.05 0.98 0.56 0.40 0.50 0.22 0.53 0.64 0.62 0.38 0.92 0.86 0.34 0.29 0.60 0.40 0.61 0.57 0.55 0.18 0.94 0.70 0.71 0.28 0.74 0.80 0.15 0.52 0.82 0.12 0.68 0.05 0.35 0.73 0.62 0.73 0.80 0.90 0.90 0.66 0.53 0.67 0.37 0.72 0.41 0.22 0.34 0.14 0.54 0.60 0.45 0.95 0.20 0.60 0.24 0.35 0.98 0.27 0.95 0.84 0.69 0.88 0.21 0.72 0.45 0.06 0.58 0.66 0.95 0.67 0.12 0.14 0.89 0.94 0.90 0.40 0.43 0.06 0.87 0.38 0.68 0.48 0.73 0.59 0.05 0.55 0.76 0.83 0.97 0.15 0.41 0.63 0.99 0.62 0.65 0.37 0.30 0.73 0.88 0.13 0.62 0.02 0.11 0.02 0.77 0.24 0.83 0.81 0.05 0.58 0.29 0.06 0.70 0.43 0.44 0.91 0.34 0.18 0.40 0.50 0.20 0.03 0.67 0.08 0.72 0.83 0.30 0.80 0.66 0.83 0.75 0.49 0.72 0.45 0.66 0.16 0.35 0.61 0.40 0.75 0.24 0.77 0.84 0.88 0.72 0.65 0.12 0.32 0.52 0.52 0.83 0.81 0.30 0.93 0.32 0.35 0.88 0.52 0.41 0.30 0.56 0.86 0.40 0.38 0.68 0.11 0.55 0.45 0.58 0.37 0.28 0.01 0.16 0.10 0.39 0.62 0.53 0.18 0.98 0.96 0.07 0.94 0.72 0.54 0.56 0.91 0.36 0.58 0.41 0.10 0.55 0.04 0.92 0.83 0.28 0.10 0.69 0.11 0.53 0.60 0.49 0.33 0.97 0.80 0.85 0.90 0.15 0.43 0.52 0.28 0.75 0.19 0.62 0.19 0.29 0.37 0.83 0.63 0.84 0.14 0.25 0.58 0.90 0.36 0.67 0.54 0.59 0.39 0.86 0.73 0.

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