hw6-sol - AMS 311 (Fall, 2010) Joe Mitchell PROBABILITY...

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Unformatted text preview: AMS 311 (Fall, 2010) Joe Mitchell PROBABILITY THEORY Homework Set # 6 Solution Notes (1). (20 points) Two fair dice are rolled. Find the joint probability mass function of X and Y , where X is the larger of the two values and and Y is the smaller of the two values on the dice. (e.g., if the roll is (4,2), then X = 4 and Y = 2 , while if the roll is (4,4), then X = Y = 4 ) Let X be the larger of the two values, and let Y be the smaller of the two values. Then, X { 1 , 2 ,... , 6 } and Y { 1 , 2 ,... , 6 } . p (1 , 1) = P ( X = 1 ,Y = 1) = P ( { (1 , 1) } ) = 1 / 36, where (1,1) is the outcome in which the first die is a 1 and the second die is also a 1 (so that the larger die is 1 and the smaller is also 1). Similarly, p ( i,i ) = P ( X = i,Y = i ) = P ( { ( i,i ) } ) = 1 / 36, for i = 1 , 2 ,... , 6. in which the first die is a 1 and the second die is also a 1 p (1 , 2) = p (1 , 3) = = p (1 , 6) = 0, since the smaller number cannot be greater than the larger of the two numbers. Similarly, p ( i,j ) = 0 for any i < j . p (2 , 1) = P ( { (1 , 2) , (2 , 1) } ) = 2 / 36. Similarly, p ( i,j ) = P ( { ( i,j ) , ( j,i ) } ) = 2 / 36 for any pair ( i,j ) with i > j and i,j { 1 , 2 ,... , 6 } . Thus, in summary, p ( i,j ) = if i < j , and i,j { 1 , 2 ,... , 6 } 1 36 if ( i,j ) { (1 , 1) , (2 , 2) ,... , (6 , 6) } 2 36 if i > j , and i,j { 1 , 2 ,... , 6 } (It is convenient to arrange all these numbers in a table.) (2). (25 points) Let X and Y be independent exponential random variables with respective parameters 2 and 3. Find the cdf and density of Z = X/Y . Also, compute P ( X < Y ) . Since X and Y are independent exponential random variables, their joint density, f ( x,y ), is given by the product of the marginals: f ( x,y ) = f X ( x ) f Y ( y ) = braceleftbigg 2 e- 2 x 3 e- 3 y if x 0, y otherwise We compute the cdf of Z = X/Y directly from the definition: F Z ( z ) = P ( Z z ) = P ( X/Y z ) = braceleftbigg if z P ( Y (1 /z ) X ) = integraltext integraltext x/z 2 e- 2 x 3 e- 3 y dydx = 2 z 2 z +3 if z > Details of the integration: integraldisplay integraldisplay x/z 2 e- 2 x 3 e- 3 y dydx = integraldisplay 2 e- 2 x bracketleftBig e- 3 y | y = x/z bracketrightBig dx = integraldisplay 2 e- 2 x e- 3 x/z dx = 2 z 2 z + 3 We get the density, f Z ( z ), by taking the derivative, case by case: f Z ( z ) = dF Z ( z ) dz = braceleftbigg if z 6 (2 z +3) 2 if z > We are also to compute P ( X < Y ) = integraldisplay integraldisplay x 2 e- 2 x 3 e- 3 y dydx = 2 2 + 3 . (In order to set up the limits of integration, draw a picture of the support set (the first quadrant of the plane) and the intersection with the region where y > x (the cone of points making an angle of 45-90 degrees with the x-axis).) More generally, if X is exponential( 1 ) and Y is exponential( 2 ), and they are independent, then P ( X < Y...
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hw6-sol - AMS 311 (Fall, 2010) Joe Mitchell PROBABILITY...

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