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handout2 - 1 18.338J/16.394J: The Mathematics of Infinite...

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Unformatted text preview: 1 18.338J/16.394J: The Mathematics of Infinite Random Matrices Histogramming Professor Alan Edelman Handout #2, Tuesday, September 14, 2004 Random Variables and Probability Densities We assume that the reader is familiar with the most basic of facts concerning continuous random variables or is willing to settle for the following sketchy description. Samples from a (univariate or multivariate) experiment can be histogrammed either in practice or as a thought experiment. Histogramming counts how many samples fall in a certain interval. Underlying is the notion that there is a probability distribution which precisely represents the probability of falling into an interval. If x R is a real random variable with probability density p x ( t ), this means that the probability that x b may be found in an interval [ a, b ] is a p x ( t )d t . More generally if S is some subset of R , the probability that x S is S p ( t )d t . Later on, we may be more careful and talk about sets that are Lebesgue measurable, but this will do for now. The probability density is roughly the picture you would obtain if you collected many random values of x and then histogrammed these values. The only problem is that if you have N samples and your bins have size , then the total area under the boxes is...
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handout2 - 1 18.338J/16.394J: The Mathematics of Infinite...

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