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Variance of Individual Prediction

We want to predict an individual Y corresponding to X = X0; that is, we want to obtain

We predict this as

Y0 — Y0 = fa1 + fa2 X0 + U0 — 01 + fa X0) = (fa1 — fa) + (fa2 — fa) X0 + U0

Therefore,

E(Y0 — Y0) = E(fa1 — fa) + E(fa2 — fa)X0 — E(u0) =0

because fa, fa are unbiased, X0 is a fixed number, and E(u0) is zero by assumption.

Squaring (7) on both sides and taking expectations, we get var (Y0 — Y0) = var (fa) + X2 var (fa) + 2X0 cov(fa1, fa2) + var(uo). Using the variance and co-variance formulas for fa and fa given earlier, and noting that var(u0) = a2, we obtain var (Y — Y0) = a2

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