27p. Gaussian random numbers _printable_

27p. Gaussian random numbers _printable_ - Gaussian Random...

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1 ©2009 by L. Lagerstrom Gaussian Random Numbers The Gaussion, or normal, distribution Creating a normal distribution using randn Simulating a noisy signal ©2009 by L. Lagerstrom The Gaussian, or Normal, Distribution In the previous lesson we learned how Matlab generates a uniform distribution of random numbers between 0 and 1. Remember that "uniform distribution" means that any number between 0 and 1 is equally likely to be generated. There are many other types of random number distributions, however. One particularly important distribution (mentioned previously in the Statistics learning module) is known as a Gaussian distribution, or "normal" distribution. This type of distribution is a bell-shaped curve, as shown below. -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.05 0.1 0.15 ©2009 by L. Lagerstrom The Normal Distribution, cont. The exact mathematical formula for a normal distribution is: where μ is the mean value (the center point) and σ is the standard deviation (a measure of the curve's width or spread). For the curve shown above, the mean is 0 and the standard deviation is 3. -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.05 0.1 0.15 2 2 2 / ) ( 2 1 ) ( σ μ σ π - - = x e x f ©2009 by L. Lagerstrom The Normal Distribution, cont. Many phenomena follow a normal distribution, or at least come close to it. For example, if we were to plot a histogram of exam scores for a large class, we would often find that it approximates the normal distribution curve: 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 Gaussian/Normal Distribution Exam score (mean = 60, std = 15) Number of scores
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2 ©2009 by L. Lagerstrom The Normal Distribution, cont. When a frequency distribution (histogram) approximates a normal distribution, it means that approximately 68% of the counted results fall within one standard deviation of the mean. In the example of exam scores on the previous slide, where the mean was 60 and the standard deviation 15, this means that about 2/3 of all the students scored between 60-15 = 45 and 60+15 = 75. In addition, it means that 1/6 of the students scored above 75 and 1/6 of the students scored below 45. (All this assumes, of course, that the distribution of exam scores did indeed follow a normal distribution curve. Sometimes, of course, the scores will be skewed one way or the other.) ©2009 by L. Lagerstrom Creating a Normal Distribution In the previous lesson we used the rand function to create a uniform distribution of random numbers. To create a normal or Gaussian distribution of random numbers in Matlab, we use the randn function (with the extra "n" representing "normal"). It works very similarly to the rand function.
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