Suppose thatnbspx1 x2 is a sequence of random variables, Basic Statistics

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Suppose thatnbspx1 x2 is a sequence of random variables

Suppose that X₁, X₂, . . .is a sequence of random variables with E(X_t²) 8 and E(X_t)=µ.
a. If X₁, X₂, . . .is iid with E(X_i²) 8 and EX_i=µ, where µis known, whatis the minimum mean squared error predictor of X_n₊₁in terms of X₁, . . . , X_n?
b. Under the conditions of part (c) show that the best linear unbiased estimator of µin terms of X₁, . . . , X_nis X¯ =(X₁ + · · · + X_n)/n. (µˆ said to be anunbiased estimator of µif Euˆ=µfor all µ.)
c. Under the conditions of part (c) show that X¯ is the best linear predictor of X_n₊₁ that is unbiased for µ.
d. If X₁, X₂, . . .is iid with E(X_i²) 8 and EX_i=µ, and if S₀₌0, S_n=X₁ + · · · + X_n, n=1,2, . . ., what is the minimum mean squared error predictor of S_n₊₁in terms of S₁, . . . , S_n?

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Basic Statistics: Suppose thatnbspx1 x2 is a sequence of random variables

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