Suppose that a random variable X can take only the five values x = 1, 2, 3, 4, 5 with the following probabilities:
f (1|θ) = θ3, f(2|θ) = θ2(1− θ),
f (3|θ) = 2θ(1− θ), f (4|θ) = θ(1− θ)2,
f (5|θ) = (1− θ)3.
Here, the value of the parameter θ is unknown (0 ≤ θ ≤ 1).
a. Verify that the sum of the five given probabilities is 1 for every value of θ.
b. Consider an estimator δc(X) that has the following form:
δc(1) = 1, δc(2) = 2 − 2c, δc(3) = c,
δc(4) = 1− 2c, δc(5) = 0.
Show that for each constant c, δc(X) is an unbiased estimator of θ.
c. Let θ0 be a number such that 0 < θ0 < 1. Determine a constant c0 such that when θ = θ0, the variance of δc0(X) is smaller than the variance of δc(X) for every other value of c.