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JOINT DISTRIBUTIONS 181 TABLE 5.5 xly 3 4 0. 0. 0. 1/35 1 12/35 0. 0. 18/35 010 4/35 0. 2. In conducting an experiment in the laboratory, temperature gauges are to be used at four junction points in the equipment setup. These four gauges are ran- domly selected from a bin containing seven such gauges. Unknown to the sci- entist, three of the seven gauges give improper temperature readings. Let X denote the number of defective gauges selected and Y the number of nondefec- tive gauges selected. The joint density for (X, Y) is given in Table 5.5. o (a) The values given in Table 5.5 can be derived by realizing that the random variable X is hypergeometric. Use the results of Sec. 3.7 to verify the val- ues given in Table 5.5. (b) Find the marginal densities for both X and Y. What type of random variable is Y? (c) Intuitively speaking, are X and Y independent? Justify your answer mathe- matically. 3. The joint density for (X, Y) is given by fxy (x, y) = 1/n² x = 1, 2, 3, . . .,n y = 1, 2, 3, . .., n (a) Verify that fyy (x, y) satisfies the conditions necessary to be a density. (b) Find the marginal densities for X and Y. (c) Are X and Y independent? . The joint density for (X, Y) is given by

JOINT DISTRIBUTIONS 181 TABLE 5.5 xly 3 4 0. 0. 0. 1/35 1 12/35 0. 0. 18/35 010 4/35 0. 2. In conducting an experiment in the laboratory, temperature gauges are to be used at four junction points in the equipment setup. These four gauges are ran- domly selected from a bin containing seven such gauges. Unknown to the sci- entist, three of the seven gauges give improper temperature readings. Let X denote the number of defective gauges selected and Y the number of nondefec- tive gauges selected. The joint density for (X, Y) is given in Table 5.5. o (a) The values given in Table 5.5 can be derived by realizing that the random variable X is hypergeometric. Use the results of Sec. 3.7 to verify the val- ues given in Table 5.5. (b) Find the marginal densities for both X and Y. What type of random variable is Y? (c) Intuitively speaking, are X and Y independent? Justify your answer mathe- matically. 3. The joint density for (X, Y) is given by fxy (x, y) = 1/n² x = 1, 2, 3, . . .,n y = 1, 2, 3, . .., n (a) Verify that fyy (x, y) satisfies the conditions necessary to be a density. (b) Find the marginal densities for X and Y. (c) Are X and Y independent? . The joint density for (X, Y) is given by

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 22E
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JOINT DISTRIBUTIONS 181 TABLE 5.5 xly 3 4 0. 0. 0. 1/35 1 12/35 0. 0. 18/35 010 4/35 0. 2. In conducting an experiment in the laboratory, temperature gauges are to be used at four junction points in the equipment setup. These four gauges are ran- domly selected from a bin containing seven such gauges. Unknown to the sci- entist, three of the seven gauges give improper temperature readings. Let X denote the number of defective gauges selected and Y the number of nondefec- tive gauges selected. The joint density for (X, Y) is given in Table 5.5. o (a) The values given in Table 5.5 can be derived by realizing that the random variable X is hypergeometric. Use the results of Sec. 3.7 to verify the val- ues given in Table 5.5. (b) Find the marginal densities for both X and Y. What type of random variable is Y? (c) Intuitively speaking, are X and Y independent? Justify your answer mathe- matically. 3. The joint density for (X, Y) is given by fxy (x, y) = 1/n² x = 1, 2, 3, . . .,n y = 1, 2, 3, . .., n (a) Verify that fyy (x, y) satisfies the conditions necessary to be a density. (b) Find the marginal densities for X and Y. (c) Are X and Y independent? . The joint density for (X, Y) is given by <r<n. na positive integer 2. 00000 23
Transcribed Image Text:JOINT DISTRIBUTIONS 181 TABLE 5.5 xly 3 4 0. 0. 0. 1/35 1 12/35 0. 0. 18/35 010 4/35 0. 2. In conducting an experiment in the laboratory, temperature gauges are to be used at four junction points in the equipment setup. These four gauges are ran- domly selected from a bin containing seven such gauges. Unknown to the sci- entist, three of the seven gauges give improper temperature readings. Let X denote the number of defective gauges selected and Y the number of nondefec- tive gauges selected. The joint density for (X, Y) is given in Table 5.5. o (a) The values given in Table 5.5 can be derived by realizing that the random variable X is hypergeometric. Use the results of Sec. 3.7 to verify the val- ues given in Table 5.5. (b) Find the marginal densities for both X and Y. What type of random variable is Y? (c) Intuitively speaking, are X and Y independent? Justify your answer mathe- matically. 3. The joint density for (X, Y) is given by fxy (x, y) = 1/n² x = 1, 2, 3, . . .,n y = 1, 2, 3, . .., n (a) Verify that fyy (x, y) satisfies the conditions necessary to be a density. (b) Find the marginal densities for X and Y. (c) Are X and Y independent? . The joint density for (X, Y) is given by <r<n. na positive integer 2. 00000 23
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