For a standard normal random variable Z N(0,1), let z be such that P(Z > z) = .

Question

For a standard normal random variable Z N(0,1), let z be such that P(Z > z) = . Then for = 5% = 0.05, z 1.645, and for = 2.5% = 0.025, z 1.96. (1) Suppose X1, X2 and X3 are independent random variables with E(Xi) = µ, V (Xi) = 2, i = 1,2,3. Let X = X1 2X2 +3X3. Find E(X) and V (X). (2) Suppose the data x = (1,2,3) is a random sample from a population with expected value µ and variance 2. Use x to estimate µ and variance 2. (3) A list of n data items has the form x = 1 n1, 1 n1,..., 1 n1,1. Find E(x) and V (x).

Explanation / Answer

X = X1 2X2 +3X3

E(X) = E(X1 2X2 +3X3) = 0    as E(Xi) = 0 and expectation is linear

Var(X) = Var(X1 2X2 +3X3) = 1 + 4 + 9

= 14

sample mean = 1/3 ( 1 + 2 +3) = 2   {estimate for population mean}

sample variance = 1     {estimate for population variance}

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