251y0032 11 28 00 ECO251 QBA1 THIRD EXAM NOVEMBER 21 2000 Name key Section Enrolled MWF TR 10 11 12 30 Part I Do all the Following 10 Points Show your work Please indicate clearly what sections of the problem you are answering If you are following a rule like E ax aE x please state it If you are using a formula state it If you answer a yes or no question explain why Assume that the following joint probability table corresponds to two variables x y a Are x and b Compute y x 0 2 4 0 25 10 05 2 05 05 15 and y P y 40 35 25 independent Why 2 the standard deviation of xy the covariance of d Compute xy the correlation of c Compute 1 10 20 05 x x x x 2 and and y y and interpret it 3 and interpret it 2 P x y 2 2 x y can take the following values e Find f 0 1 2 3 4 5 and 6 Write out the distribution of by making a list with these values and their probabilities 2 i Find the mean and standard deviation of x y from the table you just made 2 g x y ii Show that they are the same as the values of the mean and standard deviation of x y that you would have gotten using the method you used in the last part of Graded Assignment 2 2 Solution a Check for independence First you need to find P x and P y Look at the upper left hand probability below Its value is 25 and it represents P x 0 y 0 If x and y were independent we would have P x 0 y 0 P x 0 P y 0 40 40 16 Since this is not true we can say that x and y are not independent x 0 1 2 P y 0 25 10 05 40 y 2 10 20 05 35 4 05 05 15 25 P x 40 35 25 1 0 xP x 0 00 0 35 0 50 0 85 x 2 P x 0 00 0 35 1 00 1 35 P x 1 To summarize xP x 0 85 yP y y P y 0 00 0 00 0 70 1 40 x E x x2 P x 1 35 E x 2 2 P y 1 y E y E y 2 yP y 1 7 and 2 y P y 5 4 1 00 4 00 1 70 5 40 251y0032 11 28 00 E xy 25 0 0 xyP xy 10 0 2 05 0 4 10 1 0 20 1 2 05 1 4 05 2 0 0 0 05 2 2 0 0 15 2 4 0 0 0 0 0 4 0 2 0 0 0 2 2 0 1 2 xy Cov xy E xy x y 2 0 0 85 1 7 0 555 x2 E x 2 x2 1 35 0 85 2 0 6275 and 2 y E y 2 2y 5 4 1 7 2 2 51 So that xy xy x y 0 555 0 6275 2 51 0 44223 x 0 79215 y 1 5843 b Compute x the standard deviation of x 2 Above it says x2 E x 2 x2 1 35 0 85 2 0 6275 and x 0 79215 c Compute xy the covariance of x and y and interpret it 3 Above it says xy Cov xy E xy x y 2 0 0 85 1 7 0 555 Since this is positive x and y seem to move together if one rises the other also tends to rise d Compute xy the correlation of x and y and interpret it 2 Above it says xy 0 555 xy 0 44223 If we square this correlation we get 1956 Though the x y 0 6275 2 51 positive sign indicates that x and y seem to move together 1956 is quite small on a zero to one scale so the correlation is rather weak 251y0032 11 28 00 e Find P x y 2 1 the original table is copied below along with a second table with the sums of x 1 10 20 05 0 2 0 25 05 x and y y 2 10 05 y 4 05 15 get 2 by adding x 0 and y 2 or by adding x 2 0 2 4 0 0 2 4 x 1 1 3 5 2 2 4 6 There are two ways to and y 0 The joint probability of the first pair is 10 and the joint probability of the second pair is 05 so the total probability is P x y 2 10 05 15 f g x y can take the following values 0 1 2 3 4 5 and 6 Write out the distribution of x y by making a list with these values and their probabilities 2 i Find the mean and standard deviation of x y from the table you just made 2 ii Show that they are the same as the values of the mean and standard deviation of x y that you would have gotten using the method you used in the last part of Graded Assignment 2 2 We put the probabilities together for each value of x y in the same way we did P x y 4 The table follows x y P x y x y P x y x y 2 P x y 0 25 0 0 1 10 10 10 2 15 30 60 3 20 60 1 80 4 10 40 1 60 5 05 25 1 25 6 15 90 5 40 total 1 00 2 55 10 75 So E x y 2 55 and E x y 2 10 75 This means that Var x y E x y 2 E x y 2 10 75 2 55 2 4 2475 and x y 4 2475 2 0609 If we compute E x y and Var x y from the results of b d we get exactly the same results as we did above E x y E x E y x y 0 85 1 7 2 55 and Var x y x2 y2 2 xy Var x Var y 2Cov x y 0 6275 2 51 2 0 555 4 2475 4 251y0032 11 28 00 Part II Do the following problems Do at least 40 points but remember anything beyond 40 points is extra credit Show your work Please indicate clearly what sections of a problem you are answering And what columns you found table values in If you are following a rule like E ax aE x please state it If you are using a formula state it If you answer a yes or no question explain why You must do part a of problem 3 3 point penalty 1 Describe the meaning and give the probabilities for the problems below Example For the Hypergeometric Distribution P 1 x 2 when …
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