Lecture 10_notes.pdf - Econ 41(Summer 2017 Department of...

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Econ 41 ( Summer 201 7 ) Department of Economics, UCLA Instructor: Ming Gu Lecture 10: Continuous Probability Distributions Remark 1 Skip Example 3.2-5 De°nition 2 A random variable X is continuous if its support S is a continuous subset of R . De°nition 3 PDF (probability density function) of a continuous random variable X is f ( x ) such that (a) f ( x ) ° 0 for x 2 S and f ( x ) = 0 for x = 2 S ; (b) R + 1 °1 f ( x ) dx = 1; (c) Pr ( a < X < b ) = R b a f ( x ) dx ; (d) Pr ( X 2 A ) = R A f ( x ) dx: Example 4 There is a balanced spinner such that the result of a spin is a random variable X which can take values in S = f x : 0 ± x ± 1 g . It has the PDF f ( x ) = 1 for 0 ± x ± 1 . We say that such a random variable has a uniform distribution. We can see that Pr [0 : 25 < X < 0 : 75] = Z 0 : 75 0 : 25 1 dx = [ x ] 0 : 75 0 : 25 = 0 : 5; Pr [0 : 5 < X < 0 : 6] = Z 0 : 6 0 : 5 1 dx = [ x ] 0 : 6 0 : 5 = 0 : 1; Pr [0 : 1 < X < 0 : 4] = Z 0 : 4 0 : 1 1 dx = [ x ] 0 : 4 0 : 1 = 0 : 3; Pr [ X = 0 : 5] = Z 0 : 5 0 : 5 1 dx = [ x ] 0 : 5 0 : 5 = 0 : Remark 5 Uniform distribution is formally de°ned in Section 3.3 of the textbook. You are encouraged to read the discussion above Figure 3.3-1 as well as Example 3.3.1. You do not
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