1. Consider a random sample X1 , X2 , . . . , Xn from the pdf
(
0.5(1 + θx), −1 ≤ x ≤ 1
f (x; θ) =
0,
otherwise
where −1 ≤ θ ≤ 1 (this distribution arises in particle physics). Show that
θ̂ = 3X̄ is an unbiased estimator of θ. (Hint: First determine E[X̄].)
2. Suppose the true average growth µ of one type of plant during a 1-year
period is identical to that of a second type, but the variance of growth for the first
type is σ 2 , whereas for the second type, the variance is 4σ 2 . Let X1 , X2 , . . . , Xm
be m independent growth observations on the first type (so E[Xi ] = µ, V [Xi ] =
σ 2 ), and let Y1 , Y2 , . . . , Yn be n independent growth observations on the second
type (so E[Yi ] = µ, V [Yi ] = 4σ 2 ).
a. Show that for any δ between 0 and 1, the estimator µ̂ = δ X̄ + (1 − δ)Ȳ is
unbiased for µ.
b. For fixed m and n, compute V [µ̂], and then find the value of δ that minimizes V [µ̂]. (Hint: Differentiate V [µ̂] with respect to δ.)
1