12 note that the variance is proportional to the

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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 (...
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