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 continuoustype 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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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.
 Spring '08
 Zahrn
 The Land

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