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FM5002-HW9-4.11.12

# FM5002-HW9-4.11.12 - Financial Mathem atics 5002 Hom ework...

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Financial Mathematics 5002 : Homework 9 (0049) Due on 11 April 2012 Scot Adams Solutions 0049 − 1 . Define PCRVs X and Y by X L w R E 2 if 0 LΩL 0.3 3 if 0.3 lΩL 0.65 7 if 0.65 1 Y L w R E 8 if 0 0.45 9 if 0.45 1 Compute E[X | Y]. E l X Y r E E l X Y E 8 r if 0 0.45 E l X Y E 9 r if 0.45 1 E 2 S .3 P 3 S .15 .45 if 0 0.45 3 S .2 P 7 S .35 .55 if 0.45 1 E 7 3 if 0 0.45 61 11 if 0.45 1 A 2.333 if 0 0.45 5.545 if 0.45 1 . 2 . Define PCRVs U and V by U L w R E 2 if 0 0.3 3 if 0.3 0.65 7 if 0.65 1 V L w R E 800 if 0 0.45 900 if 0.45 1 Compute E[U | V]. Using 0049 M 1, we compute that E l X Y r E E l X Y E 600 r if 0 0.45 E l X Y E 700 r if 0.45 1 E 7 3 if 0 0.45 if 0.45 1 A 2.333 if 0 0.45 5.545 if 0.45 1 . 3 . a. Show the graph of some PCRV X such that Pr[X = 3] = 0.2 and Pr[X = 5] = 0.8. b. Show the graph of some PCRV Y such that X and Y are independent, Pr[Y = -1] = 0.5 and Pr[Y = -2] = 0.5. c. Show the graph of E[X | Y].

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d. Show the graph of E[Y | X]. e. Find the partition P of X. f. Find the partition Q of Y. g. Is X P - measurable? h. Is Y P - measurable? i. Is E[Y | X] P - measurable? j. Find Cov[X, Y].
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