Example a19 geometric distribution starting with the

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Unformatted text preview: next formula. If a < b, then {X b} = {X a} [ {a < X b} with the two sets on the right-hand side disjoint so P (X b) = P (X a) + P (a < X b) or, rearranging, P (a < X b) = P (X b) P (X a) = F (b) F (a) (A.10) The last formula is valid for any random variable. When X has density function f , it says that Zb f (x) dx = F (b) F (a) a i.e., the integral can be evaluated by taking the di↵erence of the antiderivative at the two endpoints. To see what distribution functions look like, and to explain the use of (A.10), we return to our examples. 213 A.2. RANDOM VARIABLES, DISTRIBUTIONS Example A.14. Uniform distribution. f (x) = 1/(b a) for a < x < b. 8 >0 xa < F (x) = (x a)/(b a) a x b > : 1 xb To check this, note that P (a < X < b) = 1 so P (X x) = 1 when x P (X x) = 0 when x a. For a x b we compute Zx Zx 1 xa P (X x) = f (y ) dy = dy = ba ba 1 a b and In the most important special case a = 0, b = 1 we have F (x) = x for 0 x 1. Example A.15. Exponential distribution. f (x) = e ( 0 x0 F (x) = 1exx0 x for x 0. The firs...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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