5-5 COVARIANCE AND CORRELATION

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EXERCISES FOR SECTION 5 4

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5-55. Suppose the random variables X, Y, and Z have the joint probability density function f 1x, y, z2 8xyz for 0 x 1, 0 y 1, and 0 z 1. Determine the following: (a) P1X 0.52 (b) P1X 0.5, Y 0.52 (c) P1Z 22 (d) P1X 0.5 or Z 22 (e) E1X 2 5-56. Continuation of Exercise 5-55. Determine the following: (a) P1X 0.5 Y 0.52 (b) P1X 0.5, Y 0.5 Z 0.82 5-57. Continuation of Exercise 5-55. Determine the following: (a) Conditional probability distribution of X given that Y 0.5 and Z 0.8 (b) P1X 0.5 Y 0.5, Z 0.82 5-58. Suppose the random variables X, Y, and Z have the joint probability density function fXYZ (x, y, z) c over the cylinder x2 y2 4 and 0 z 4. Determine the following. (a) The constant c so that fXYZ (x, y, z) is a probability density function (b) P1X 2 Y 2 22 (c) P1Z 22 (d) E1X 2 5-59. Continuation of Exercise 5-58. Determine the following: (a) P1X 1 Y 12 (b) P1X 2 Y 2 1 Z 12 5-60. Continuation of Exercise 5-58. Determine the conditional probability distribution of Z given that X 1 and Y 1. 5-61. Determine the value of c that makes fXYZ (x, y, z) c a joint probability density function over the region x 0, y 0, z 0, and x y z 1. 5-62. Continuation of Exercise 5-61. Determine the following: (a) P1X 0.5, Y 0.5, Z 0.52 (b) P1X 0.5, Y 0.52 (c) P1X 0.52 (d) E1X 2 5-63. Continuation of Exercise 5-61. Determine the following: (a) Marginal distribution of X (b) Joint distribution of X and Y (c) Conditional probability distribution of X given that Y 0.5 and Z 0.5 (d) Conditional probability distribution of X given that Y 0.5 5-64. The yield in pounds from a day s production is normally distributed with a mean of 1500 pounds and standard deviation of 100 pounds. Assume that the yields on different days are independent random variables. (a) What is the probability that the production yield exceeds 1400 pounds on each of ve days next week (b) What is the probability that the production yield exceeds 1400 pounds on at least four of the ve days next week 5-65. The weights of adobe bricks used for construction are normally distributed with a mean of 3 pounds and a standard deviation of 0.25 pound. Assume that the weights of the bricks are independent and that a random sample of 20 bricks is selected. (a) What is the probability that all the bricks in the sample exceed 2.75 pounds (b) What is the probability that the heaviest brick in the sample exceeds 3.75 pounds 5-66. A manufacturer of electroluminescent lamps knows that the amount of luminescent ink deposited on one of its products is normally distributed with a mean of 1.2 grams and a standard deviation of 0.03 grams. Any lamp with less than 1.14 grams of luminescent ink will fail to meet customer s specifications. A random sample of 25 lamps is collected and the mass of luminescent ink on each is measured. (a) What is the probability that at least 1 lamp fails to meet speci cations (b) What is the probability that 5 lamps or fewer fail to meet speci cations (c) What is the probability that all lamps conform to speci cations (d) Why is the joint probability distribution of the 25 lamps not needed to answer the previous questions

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COVARIANCE AND CORRELATION

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When two or more random variables are de ned on a probability space, it is useful to describe how they vary together; that is, it is useful to measure the relationship between the variables. A common measure of the relationship between two random variables is the covariance. To de ne the covariance, we need to describe the expected value of a function of two random variables h(X, Y ). The de nition simply extends that used for a function of a single random variable.

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