12 note that the variance is proportional to the

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: e axiom that probability is additive, Axiom P.2, only holds for countably infinite sums. If a < b then b P {a < X ≤ b} = FX (b) − FX (a) = fX (u)du. a Since P {X = a} = P {X = b} = 0, it follows more generally that: b P {a < X ≤ b} = P {a < X < b} = P {a ≤ X ≤ b} = P {a ≤ X < b} = fX (u)du. a So when we work with continuous-type random variables, we don’t have to be precise about whether the endpoints of intervals are included when calculating probabilities. It follows that the integral of fX over every interval (a, b) is greater than or equal to zero, so fX must be a nonnegative function. Also, ∞ 1 = lim lim FX (b) − FX (a) = a→−∞ b→+∞ fX (u)du. −∞ Therefore, fX integrates to one. In most applications, the density functions fX are continuous, or piecewise continuous. Although P {X = u} = 0 for any real value of u, there is still a fairly direct interpretation of fX involving probabilities. Suppose uo is a constant such that fX is continuous at uo . Then for > 0, P uo − 2 < X < uo + 2 = uo + 2 fX (v )dv uo − 2 = uo + 2 fX (...
View Full Document

This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

Ask a homework question - tutors are online