(a) Recalling the definition of sigma^2 for a single rv X, write a formula that
ID: 3219737 • Letter: #
Question
(a) Recalling the definition of sigma^2 for a single rv X, write a formula that would be appropriate for computing the variance of a function h(X, Y) of two random variables. V[h(X, Y)] = E{[h(X, Y) - E(h(X, Y))]^2} = Sigma_x Sigma_y[h(x, ) - E(h(X, Y))]^2 middot p(, y) (b) An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y= the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table. 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. () V[max(X, Y)] = ____Explanation / Answer
h(x,y) =
max(x,y) probability
0 0.02 = 0.02
5 0.06+0.04 +0.17 =0.27
10 0.02+0.20+0.12+0.01+0.15 = 0.5
15 0.10+0.10+0.01 =0.21
Var(h(x,y) = E( Z^2) - E(Z)^2 ,where Z = max(x,y)
E(Z) = 0*0.02 +5* 0.27 +10* 0.5 +15*0.21 = 9.5
E(Z^2) = 0^2*0.02 +5^2* 0.27 +10^2* 0.5 +15^2*0.21 = 104
hence variance = 104 - 9.5^2 = 13.75
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