16 C H A P T E R 2. P R O B A B I L I T Y A N D S T A T I S T I C S E x a m p l e 2.8 Let f~ be the set of all objects in Figure 2.1, and assign each object a probability of 1/13. Let A be the set os all objects containing an "A," B be the set of all objects containing a "B," and Black be the set of all black objects. Then according to Bayes' Theorem, P(AIBl~k)P(Bl~r P(Black[A) = P(A[Black)P(Black)+ P(A[White)P(White) (9)+ which i~ th~ ~ ~ .~lu~ we g~t by computing 3 5' P(B,~r directly. In the previous example we can just as easily compute P(BlacklA ) directly. We will see a useful application of Bayes' Theorem in Section 2.4. 2.2 Random Variables In this section we present the formal definition and mathematical properties of a random variable. In Section 2.4 we show how they are developed in practice. 2.2.1 P r o b a b i l i t y D i s t r i b u t i o n s of R a n d o m Variables Definition 2.5 Given a probability space (f~, P), a r a n d o m variable X is a function whose domain is f~. The range of X is called the space of X. E x a m p l e 2.9 Let f~ contain all outcomes of a throw os a pair os six-sided dice, and let P assign 1/36 to each outcome. Then f~ is the following set os ordered pairs: f~= {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2),... (6, 5), (6, 6)}. Let the random variable X assign the sum of each ordered pair to that pair, and let the random variable Y assign "odd" to each pair of odd numbers and "even" to a pair is at least one number in that pair is an even number. The following table shows some of the values os X and Y: (1,1) (1, 2) (2, 1) (6,6) x(~) 2 3 Y(~) odd even 3 12 even even The space of X is {2, 3, 4, 5, 6, r, 8, 9,10,11,12}, and that of Y is {odd, even}.