Stochastic

When hx xk ehx ex k is the k th moment when hx

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Unformatted text preview: bability density. Note that the last property implies f (x) = c for a < x < b. In this case the integral is c(b a), so we must pick c = 1/(b a). Example A.12. Exponential distribution. ( exx0 f (x) = 0 otherwise Here that > 0 is a parameter. To check that this is a density function, we note Z1 1 e x dx = e x 0 = 0 ( 1) = 1 0 In a ﬁrst course in probability, the next example is the star of the show. However, it will have only a minor role here. Example A.13. Normal distribution. f (x) = (2⇡ ) 1 /2 e x2 /2 Since there is no closed form expression for the antiderivative of f , it takes some ingenuity to check that this is a probability density. Those details are not important here, so we will ignore them. Any random variable (discrete, continuous, or in between) has a distribution function deﬁned by F (x) = P (X x). If X has a density function f (x) then Zx F (x) = P ( 1 < X x) = f (y ) dy 1 That is, F is an antiderivative of f . One of the reasons for computing the distribution function is explained by the...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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