Fundamentals of Probability and Statistics for Engineers

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Random Variables and Probability Distributions 41F X (x)11218–2 –1 0 1 2 3xFigure 3.1 Probability distribution function of X, F X x), for Example 3.1F X (x)1.00.2–1 0 1xFigure 3.2Probability distribution function of a continuous random variable X, F X x)3.2.2 PROBABILITY MASS FUNCTION FOR DISCRETE RANDOMVARIABLESLet X be a discrete random variable that assumes at most a countably infinitenumber of values x 1, x2,... with nonzero probabilities. If we denotePX ˆ x i ) ˆ px i ), i ˆ 1, 2, . . ., then, clearly,90 < p…x i †1; for all i; =Xp…x i †ˆ1:…3:4†;iTLFeBOOK
42 Fundamentals of Probability and Statistics for Engineersp X (x)1218–2 –10 1 2 3 4xFigure 3.3 Probability mass function of X, p X x), for the random variable definedin Example 3.1Definition 3.2. The functionp X …x† ˆP…X ˆ x†:…3:5†is defined as the probability mass function (pmf) of X. Again, the subscript X isused to identify the associated random variable.For the random variable defined in Example 3.1, the pmf is zero everywhereexcept at x i , i ˆ 1, 2, . . . , and has the appearance shown in Figure 3.3.This is a typical shape of pmf for a discrete random variable. SincePX ˆ x) ˆ 0 for any x for continuous random variables, it does not exist inthe case of the continuous random variable. We also observe that, like F X x),the specification of p X x) completely characterizes random variable X; furthermore,these two functions are simply related by:p X …x i †ˆF X …x i † F X …x i 1 †; …3:6†i:xF X …x† ˆ X ixp X …x i †;iˆ1…3:7†(assuming x 1 < x 2 < . ..).The upper limit for the sum in Equation (3.7) means that the sum is takenover all i satisfying x i x. Hence, we see that the PDF and pmf of a discreterandom variable contain the same information; each one is recoverable fromthe other.TLFeBOOK
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390 Subject IndexExtreme-value dist
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