Ebook A first course in probability (8th edition): Part 2

(BQ) Part 2 book "A first course in probability" has contents: Jointly distributed random variables, properties of expectation, limit theorems, additional topics in probability, simulation. Invite you to reference. | C H A P T E R 6 Jointly Distributed Random Variables JOINT DISTRIBUTION FUNCTIONS INDEPENDENT RANDOM VARIABLES SUMS OF INDEPENDENT RANDOM VARIABLES CONDITIONAL DISTRIBUTIONS: DISCRETE CASE CONDITIONAL DISTRIBUTIONS: CONTINUOUS CASE ORDER STATISTICS JOINT PROBABILITY DISTRIBUTION OF FUNCTIONS OF RANDOM VARIABLES EXCHANGEABLE RANDOM VARIABLES JOINT DISTRIBUTION FUNCTIONS Thus far, we have concerned ourselves only with probability distributions for single random variables. However, we are often interested in probability statements concerning two or more random variables. In order to deal with such probabilities, we define, for any two random variables X and Y, the joint cumulative probability distribution function of X and Y by F(a, b) = P{X a, Y b} − q a, Y > b} = 1 =1 =1 =1 =1 − − − − − P({X > a, Y > b}c ) P({X > a}c ∪ {Y > b}c ) P({X a} ∪ {Y b}) () [P{X a} + P{Y b} − P{X a, Y b}] FX (a) − FY (b) + F(a, b) Equation () is a special case of the following equation, whose verification is left as an exercise: P{a1 0 Similarly, pY (y) = p(x, y) x:p(x,y)>0 EXAMPLE 1a Suppose that 3 balls are randomly selected from an urn containing 3 red, 4 white, and 5 blue balls. If we let X and Y denote, respectively, the number of red and white balls chosen, then the joint probability mass function of X and Y, p(i, j) = P{X = i, Y = j}, is given by 12 3 p(0, 0) = 5 3 p(0, 1) = 4 1 5 2 p(0, 2) = 4 2 5 1 p(0, 3) = 4 3 12 3 10 220 40 12 = 3 220 30 12 = 3 220 4 = 220 = 234 Chapter 6 Jointly Distributed Random Variables p(1, 0) = 3 1 5 2 p(1, 1) = 3 1 4 1 p(1, 2) = 3 1 4 2 12 3 p(2, 0) = 3 2 5 1 12 3 p(2, 1) = 3 2 4 1 12 3 p(3, 0) = 3 3 30 220 60 12 = 3 220 18 = 220 15 = 220 12 = 220 12 3 = 5 1 12 3 = 1 220 These probabilities can most easily be expressed in tabular form, as in Table . The reader should note that the probability mass function of X is obtained by computing the .