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20e-lecture20

# 20e-lecture20 - Lecture 20 Friday May 15th [email protected]

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Lecture 20 - Friday May 15th [email protected] Key words: Probability distribution function, Gaussian distribution Key concepts: Know how to show that f ( x ) = 1 2 π e - x 2 / 2 is a pdf. 20.1 Probability and Expectation Recall a function f : R 2 R defines a probability distribution function on R n if f ( x ) 0 for all x and Z Z · · · Z R n fdV = 1 . Underlying the probability distribution function is a probability measure defined by some experiment which gives an outcome to each point in R n . If X is the outcome of the experiment, then the probability distribution function gives P ( X λ ) = Z Z · · · Z E fdV where E = { x R n : X λ } – so E is the set of points in R n which witness the outcome of the experiment being at least λ . A basic example of a probability distribution function is the uniform distribution . Here we define f ( x, y ) = 1 for ( x, y ) D and D = [0 , 1] × [0 , 1] and f ( x, y ) = 0 otherwise. Then f ( x, y ) 0 for all ( x, y ) R 2 and RR D fdA = 1, so f is indeed a probability distribution function. Now suppose the experiment we perform is to measure the distance from a random point in D to the origin. Let X be the outcome of this experiment, so that X ( x, y ) = p x 2 + y 2 . Now consider the event X 1. The points which witness this outcome are the points ( x, y ) D such that x 2 + y 2 1. Then we see

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20e-lecture20 - Lecture 20 Friday May 15th [email protected]

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