Midterm

1. The likelihood function is

f_n is maximized if (q₂ - q₁) is minimized. Therefore the MLE of q₁ is the largest possible value min{x₁, …,x_n} and the MLE of q₂ is the smallest possible value max{x₁, …,x_n}.

2. Let . It is unbiased because

The UMVUE of is

Let Y=X₁+X₂, we have

Given Y = 2+3=5, the UMVUE = 6/2⁵

3. By N-P lemma, the most powerful test rejects H₀ if f₀(x)<kf₁(x). That is

-2x+2x < k 2x Þ x > k/(k+1) = c for a constant c.

. Therefore, the test rejects H₀ if X > 0.776

The power = Pr(X > 0.776|q = 0) = = 0.397

4. (a)

(b)

. Since , it is efficient.

5. (a) .

Let x(q|x) = c e^-q, we have

Therefore, the posterior pdf is

(b) Bayes estimator is the posterior mean =

6. (a) Y_n has pdf

Pr( Reject H₀|q=1) = Pr(Y_n > 0.99|q=1) =

(b) Pr(Y_n £ 0.99|q=2) = =0.245