Formula a16 generalizes in the usual way to

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Unformatted text preview: t this is a probability function we recall ex = 1 X xk k=0 k! (A.7) so the proposed probabilities are nonnegative and sum to 1. In many situations random variables can take any value on the real line or in a certain subset of the real line. For concrete examples, consider the height or weight of a person chosen at random or the time it takes a person to drive from Los Angeles to San Francisco. A random variable X is said to have a continuous distribution with density function f if for all a b we have Zb P (a X b) = f (x) dx (A.8) a Geometrically, P (a X b) is the area under the curve f between a and b. In order for P (a X b) to be nonnegative for all a and b and for P ( 1 < X < 1) = 1 we must have Z1 f (x) 0 and f (x) dx = 1 (A.9) 1 Any function f that satisfies (A.9) is said to be a density function. We will now define three of the most important density functions. 212 APPENDIX A. REVIEW OF PROBABILITY Example A.11. Uniform distribution on (a,b). ( 1/(b a) a < x < b f (x) = 0 otherwise The idea here is that we are picking a value “at random” from (a, b). That is, values outside the interval are impossible, and all those inside have the same pro...
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