(a) Recalling the definition of o for a single rv x, write a formula that would
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Question
(a) Recalling the definition of o for a single rv x, write a formula that would be appropriate for computing the varlance of a function hX, Y) of two random variables. [Hint: Remember that variance is just a special expected value.] hx, y (b) An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X the number of points eaned on the first part and Y the number of points earned on the second part. Suppose that the joint pmf of X and Yis given in the accompanying table. p(x, y0 510 15 0 0.02 0.06 0.02 0.10 x 5 0.04 0.13 0.20 0.10 10 0.01 0.15 0.16 0.01 Use the formula from part (a) to compute the variance of the recorded score h(X, Y)- max(X, Y) if the maximum of the two scores is recorded. (Round your answer to two decimal places.) 12.95Explanation / Answer
The distribution for h(X,Y) here is first computed as:
Therefore now first we compute the mean of h(X,Y) as:
E(max(X, Y) ) = 0*0.02 + 5*0.23 + 10*0.54 + 15*0.21 = 9.7
The second moment of h(X,Y) here is computed as:
E[ (max(X, Y) )2 ] = 0*0.02 + 52*0.23 + 102*0.54 + 152*0.21 = 107
Now the variance here is computed as:
V[ max(X,Y) ] = E[ (max(X, Y) )2 ] - [ E(max(X, Y) ) ]2 = 107 - 9.72 = 12.91
Therefore 12.91 is the required variance here.
K = h(X,Y) = Max(X,Y) P(K = k) 0 0.02 5 0.04 + 0.13 + 0.06 = 0.23 10 0.01 + 0.15 + 0.16 + 0.2 + 0.02 = 0.54 15 0.1 + 0.1 + 0.01 = 0.21Related Questions
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