Stochastic

# Formula a16 generalizes in the usual way to

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online