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Hw6 - SIEO 3600(IEOR Majors Introduction to Probability and...

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SIEO 3600 (IEOR Majors) Assignment #6 Introduction to Probability and Statistics February 24, 2010 Assignment #6 – due March 2nd, 2010 ( Expectation and variance of random variables ) 1. (continued with Problem 2 in HW5) X and Y are two continuous random variables with joint probability density function f X,Y ( x, y ) = 10 xy 2 , 0 < x < y < 1; 0 , otherwise . (a) Compute E [ X ] and E [ Y ]. ( Hint : You have calculated the marginal PDF of X and Y in HW5) (b) Compute E [ X 2 ] and E [ Y 2 ]. (c) Use (a) and (b) to compute var( X ) and var( Y ). 2. Suppose that X and Y are independent continuous random variables. Show that (a) P ( X + Y a ) = R -∞ F X ( a - y ) f Y ( y ) dy (b) P ( X Y ) = R -∞ F X ( y ) f Y ( y ) dy where f Y is the density function of Y , and F X is the distribution function of X . 3. Each night different meteorologists give us the “probability” that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p , then he or she will receive a score of 1 - (1 - p ) 2
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