1. Various Random Number Generators and the Applications
Jeff Atkinson
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2. The advantages of using random numbers
To make complicated calculation simple
To estimate the result efficiently
To simulate nature’s randomness
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3. How to find  with random numbers
Throw darts randomly to the target shown.
Count how many darts placed within the circle.
Calculate the percentage that gives the area of the circle.
Divide it by r2 to obtain . r r
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4. Case 1 The number of darts inside the shaded area = 6
The total number of darts = 8
The area of full circle = 46/8=3
If r=1, the above is also .
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5. Case 2 The number of darts inside the shaded area = 12
The total number of darts = 17
The area of full circle = 412/17=2.824
If r=1, the above is also .
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6. Case 3 The number of darts inside the shaded area = 47,000
The total number of darts = 60,000
The area of full circle = 447,000/60,000=
If r=1, the above is also .
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7. Uniformly distributed random numbers
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8. Box-Muller Distribution
All data is centered around an average value and most data stays within a certain radius called the standard deviation (σ).
To create a Box-Muller distribution, the following equations are applied to two uniformly distributed random numbers, U1 and U2:
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9. Arcsine “Bucket” Distribution
Most data comes together the upper and lower bounds. The numbers are from logistic map.
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10. Computer simulation with 10,000 uniform random numbers
The approximated  =
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11.  with Box-Muller  with different standard deviations STDV ---- 
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12. Random Walk
Mathematical determination of a path that is generated by a succession of random values.
Simulation of Brownian motion and other natural random motions
A formalism that are applied to other complex science and engineering
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13. Random Walk with Uniform Random Number
Eventual Average Distance = N
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14. Random Walk with Box-Muller
Eventual Average Distance = N
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15. Random Walk with Arcsine
Eventual Average Distance = 1.617N
N Distance .
10,
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16. Conclusions
For mathematical problems, such as , the data must be truly random and uniform.
The Box-Muller distribution does not work to find  because it depends on the standard deviation.
The random walk with arcsine distribution with logistic map [yn+1=4.0yn(1-yn)] gives the eventual average distance as 1.617N computationally.
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