Chapter 10 Continuous probability distributions - Ugrad.math.ubc.ca

Math 103 Notes Chapter 10Unlike our previous discrete probability, we will not ask “what is the probability that x takeson some exact value?” Rather, we ask for the probability that x is within some range of values,and this is computed by performing an integral. (Remark: the probability that x is exactly equalto b is the integral∫ bbp(x) dx = 0; the value is zero, by properties of the definite integral.)DefinitionThe cumulative distribution function F(x) represents the probability that the random variable takeson any value up to x, i.e.F(x) =∫ xap(s) ds.The cumulative distribution is simply the area under the probability density.The above definition has several implications:Properties of continuous probability1. Since p(x) ≥ 0, the cumulative distribution is an increasing function.2. The connection between the probability density and its cumulative distribution can be written(using the Fundamental Theorem of Calculus) as3. F(a) = 0. This follows from the fact thatp(x) = F ′ (x).F(a) =∫ aBy a property of the definite integral, this is zero.4. F(b) = 1. This follows from the fact thatap(s) ds.by property 2 of p(x).F(b) =∫ bap(s) ds = 15. The probability that x takes on a value in the interval a 1 ≤ x ≤ a 2 is the same asF(a 2 ) − F(a 1 ).This follows from the additive property of integrals:∫ a2p(s) ds −∫ a1p(s) ds =∫ a2aaa 1p(s) dsv.2005.1 - January 5, 2009 2