Suppose E(X) = 5 E[X(X - 1)] = 27.5 What is a.? ?E(X2)? [Hint: E[X(X - 1)] = E[X2 - X] = E(X2) - E(X)]? b.? ?V(X)? c.? ?The general relationship among the quantities E(X), E[X(X - 1)], and V(X)?

Problem 42E Answer: Step1: We have E(X) = 5 E[X(X - 1)] = 27.5. We need to find, 2 2 2 a. E(X ) [Hint: E[X(X - 1)] = E[X - X] = E(X ) - E(X)] b. V(X) c. The general relationship among the quantities E(X), E[X(X - 1)], and V(X) Step2: a). Consider, 2 We can write E(X ) as E[X(X - 1)] + E(X) Now, E(X ) = E[X(X - 1)] + E (X) From the given information we have E (X) = 5 E[X(X - 1)] = 27.5. Substitute above equation we get 2 E(X ) = {E[X(X - 1)] + E (X)} = 27.5 + 5 = 32.5 2 Therefore, E(X ) = 32.5. b). We can write V(X) as {E(X ) - [ E (X)] } Var(x) = {E(X ) - [ E(X)] } 2 = {32.5 - (5) }2 = {32.5 - 25} = 7.5 c). The general relationship among the quantities E(X), E[X(X - 1)], and V(X) is given below 2 Var(X) = E[X(X - 1)] + E(X) - E(X)