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Mathematical Statistics with Applications, Seventh Edition

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www.downloadslide.com 5.3 Marginal and Conditional Probability Distributions 235 5.15 The management at a fast-food outlet is interested in the joint behavior of the random variables Y 1 , defined as the total time between a customer’s arrival at the store and departure from the service window, and Y 2 , the time a customer waits in line before reaching the service window. Because Y 1 includes the time a customer waits in line, we must have Y 1 ≥ Y 2 . The relative frequency distribution of observed values of Y 1 and Y 2 can be modeled by the probability density function { e −y 1 , 0 ≤ y 2 ≤ y 1 < ∞, f (y 1 , y 2 ) = 0, elsewhere with time measured in minutes. Find a P(Y 1 < 2, Y 2 > 1). b P(Y 1 ≥ 2Y 2 ). c P(Y 1 − Y 2 ≥ 1). (Notice that Y 1 − Y 2 denotes the time spent at the service window.) 5.16 Let Y 1 and Y 2 denote the proportions of time (out of one workday) during which employees I and II, respectively, perform their assigned tasks. The joint relative frequency behavior of Y 1 and Y 2 is modeled by the density function { y1 + y 2 , 0 ≤ y 1 ≤ 1, 0 ≤ y 2 ≤ 1, f (y 1 , y 2 ) = 0, elsewhere. a Find P(Y 1 < 1/2, Y 2 > 1/4). b Find P(Y 1 + Y 2 ≤ 1). 5.17 Let (Y 1 , Y 2 ) denote the coordinates of a point chosen at random inside a unit circle whose center is at the origin. That is, Y 1 and Y 2 have a joint density function given by ⎧ ⎨ 1 f (y 1 , y 2 ) = π , y2 1 + y2 2 ≤ 1, ⎩ 0, elsewhere. Find P(Y 1 ≤ Y 2 ). 5.18 An electronic system has one each of two different types of components in joint operation. Let Y 1 and Y 2 denote the random lengths of life of the components of type I and type II, respectively. The joint density function is given by { (1/8)y1 e −(y 1+y 2 )/2 , y 1 > 0, y 2 > 0, f (y 1 , y 2 ) = 0, elsewhere. (Measurements are in hundreds of hours.) Find P(Y 1 > 1, Y 2 > 1). 5.3 Marginal and Conditional Probability Distributions Recall that the distinct values assumed by a discrete random variable represent mutually exclusive events. Similarly, for all distinct pairs of values y 1 , y 2 , the bivariate events (Y 1 = y 1 , Y 2 = y 2 ), represented by (y 1 , y 2 ), are mutually exclusive events. It follows that the univariate event (Y 1 = y 1 ) is the union of bivariate events of the type (Y 1 = y 1 , Y 2 = y 2 ), with the union being taken over all possible values for y 2 .

www.downloadslide.com 236 Chapter 5 Multivariate Probability Distributions Then For example, reconsider the die-tossing experiment of Section 5.2, where Y 1 = number of dots on the upper face of die 1, Y 2 = number of dots on the upper face of die 2. P(Y 1 = 1) = p(1, 1) + p(1, 2) + p(1, 3) +···+ p(1, 6) = 1/36 + 1/36 + 1/36 +···+1/36 = 6/36 = 1/6 P(Y 1 = 2) = p(2, 1) + p(2, 2) + p(2, 3) +···+ p(2, 6) = 1/6 . . . P(Y 1 = 6) = p(6, 1) + p(6, 2) + p(6, 3) +···+ p(6, 6) = 1/6. Expressed in summation notation, probabilities about the variable Y 1 alone are P(Y 1 = y 1 ) = p 1 (y 1 ) = 6∑ p(y 1 , y 2 ). Similarly, probabilities corresponding to values of the variable Y 2 alone are given by p 2 (y 2 ) = P(Y 2 = y 2 ) = y 2 =1 6∑ p(y 1 , y 2 ). Summation in the discrete case corresponds to integration in the continuous case, which leads us to the following definition. y 1 =1 DEFINITION 5.4 a Let Y 1 and Y 2 be jointly discrete random variables with probability function p(y 1 , y 2 ). Then the marginal probability functions of Y 1 and Y 2 , respectively, are given by p 1 (y 1 ) = ∑ p(y 1 , y 2 ) all y 2 and p 2 (y 2 ) = ∑ p(y 1 , y 2 ). all y 1 b Let Y 1 and Y 2 be jointly continuous random variables with joint density function f (y 1 , y 2 ). Then the marginal density functions of Y 1 and Y 2 , respectively, are given by f 1 (y 1 ) = ∫ ∞ −∞ f (y 1 , y 2 ) dy 2 and f 2 (y 2 ) = ∫ ∞ −∞ f (y 1 , y 2 ) dy 1 . The term marginal, as applied to the univariate probability functions of Y 1 and Y 2 , has intuitive meaning. To find p 1 (y 1 ),wesump(y 1 , y 2 ) over all values of y 2 and hence accumulate the probabilities on the y 1 axis (or margin). The discrete and continuous cases are illustrated in the following two examples.

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