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簡答：
1.(a) T
(b) F
(c) T
(d) F
(e) T
(f) F
(g) F
(h) T
(i) F
2.(a) P(Y＝y)＝(1－1/e)‧(1/e)^(y－1)
(b) Geometric(1－1/e)
(c) E(Y)＝e/(e-1)
2 2
3.(a) (1/2π)‧e^(－(x_1+x_2)/2), －∞＜x_1,x_2＜∞
2 2
(b) (1/8π)‧e^(－(w_1+w_2)/8), －∞＜w_1,w_2＜∞
___ 2
(c) f(w_1)＝(1/2√2π)‧e^(－w_1 /8) ～N(0,4)
___ 2
f(w_2)＝(1/2√2π)‧e^(－w_2 /8) ～N(0,4)
(d) since f(w_1,w_2)＝f(w_1)f(w_2)
(e) 1－e^(－1)
4.(a) (略)
︿ ___
(b) standard error of θ_1＝1/√12n
(c) (略)
︿ __________________
(d) standard error of θ_2＝√n/((n+2)‧(n+1)^2)
︿ ︿
(e) eff_n(θ_1,θ_2)＝12n^2/((n+2)‧(n+1)^2)
︿ ︿ ︿
(f) MSE(θ_i)＝Var(θ_i), i＝1,2, θ_2 is better
n r
5.(a) Σ y_1: sufficient statistic
i=1
n r
(b) (Σ y_1)/n : MLE of θ
i=1
(c) (略)
(d) Fisher information: 1/(θ^2)
Asymptotic variance of the MLE: (θ^2)/n
n
6.(a) Σ log(y_i): sufficient and complete statistic
i=1
(b) (略)
(c) (略)
n
(d) (n－1)/Σ w_i: unbiased estimator
i=1
(e) Cramer-Rao lower bound: (θ^2)/n
n
(f) Var((n－1)/Σ w_i)＝(θ^2)/(n－2): larger than the C-R lower bound
i=1
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