Question

5. Let X1, X2, . . . be independent random variables all with mean E(Xi) =...

5. Let X1, X2, . . . be independent random variables all with mean E(Xi) = 7 and variance
Var(Xi) = 9. Set
Yn =
X1 + X2 + · · · + Xn
n
(n = 1, 2, 3, . . .)
(a) Find E(Y2) and E(Y5).
(b) Find Cov(Y2, Y5).
(c) Find E (Y2 | X1).
(d) How should your answers from parts (a)–(c) be modified if the numbers “2”, “5”, “7” and
“9” are replaced by m, n, µ and σ
2
respectively?

Math   Statistics-And-Probability   
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Homework Answers

Answer #1

a) E( Y2) = E(X1 + X2 ) = E(X1) +E(X2) = 7 + 7 = 14

E(Y5) = E(X1 + X2 + X3 + X4 + X5) = 5 * 7 = 35

b) Cov( Y2 , Y5) = Cov ( X1 + X2 , (X1 + X2 + X3 + X4 + X5) = = Var(X1) + Var(X2) = 9 + 9 = 18

( since X1, X2, ... Xn , are independent).

c) E( Y2 | X1) = E( X1 + X2 | X1 ) = X1 + E(X2\X1) ( because X1 is given so its value is known).

Therefore E( Y2 | X1) = X1 + E(X2) { because X1 and X2 are independent}

Therefore E( Y2 | X1) = X1 + 7

d) E(Ym) = m*

E(Yn) = n

Cov(Ym, Yn) = min{ m, n} *

E(Ym | X1) = X1+ (m -1 )

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