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DEPARTMENT OF DECISION SCIENCES DSC3703 Study Guide 3 Mathematics and Statistics for Simulation University of South Africa Pretoria
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c 2010 University of South Africa All rights reserved. Printed and published by the University of South Africa, Muckleneuk, Pretoria. DSC3703/3/2011 Cover: Oos-Transvaal, Laeveld (1928) (”Eastern Transvaal, Lowveld”) J.H. Pierneef. J.H. Pierneef is a well-known South African artist. Permission for the use of this work was kindly granted by the Schweickerdt family. The tree structure is a recurring theme in various branches of the decision sciences.
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iii DSC3703/3/2011 Contents I Notation 1 II Number theory and combinatorics 3 1 Modular arithmetic 3 2 Prime numbers 6 3 The greatest common divisor and the least common multiple 8 4 Counting configurations 10 4.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 III Statistical distributions 17 5 Random variables 17 6 Probability density functions 20 7 Cumulative distribution functions 24 8 Distributions 27 8.1 The uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 8.2 The exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 8.3 The normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8.4 The triangular distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.4.1 The probability density function . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.4.2 Note on Winston 21.5 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . 38 8.4.3 A random number generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 IV Solutions to exercises 43
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DSC3703/3/2011 iv List of Figures 1 Illustration of Exercise 10:2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Frequency of battery lifetime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Histogram of 435 experiments with cellular telephones. . . . . . . . . . . . . . . . . . 19 4 Histogram transformed to show probabilities . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Probability density function for the experiment in Example 10:2 . . . . . . . . . . . . . 20 6 Area under the graph of f ( x ) between a and b . . . . . . . . . . . . . . . . . . . . . . . 21 7 Probability density function of the uniform distribution U [ a ; b ] . . . . . . . . . . . . . . 28 8 Probability density function of the exponential distribution with λ = 2. . . . . . . . . 31 9 Normal distribution with µ = 5 and σ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . 33 10 The standardised normal distribution n (0; 1) . . . . . . . . . . . . . . . . . . . . . . . . 34 11 The standardised cumulative normal distribution function N ( x ) . . . . . . . . . . . . . 35 12 Parameters of the triangular distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 36 13 Probability density function of the triangular distribution triang( a ; c ; b ) . . . . . . . . . 37 14 The complete graph K 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
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1 DSC3703/3 Part I Notation The equals sign (=) indicates that the two expressions to its left and right are exactly the same. To indicate two expressions that are approximately the same, we use the symbol “ ”. The sign “ ” is used to indicate equivalence. (See Section 1.) To indicate that we are defining something, we use the symbol “:=”. It states that what is to the left of := is defined to be equal to what is to the right. The symbol “ ” is used to indicate statistical distributions. For example, X U [0; 1) means that the random variable X is uniformly distributed over the interval [0; 1). The symbols x indicate the largest integer less than or equal to x , and x the smallest integer greater than or equal to x . Thus 14 3 is 4, and 8 is just 8.
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