Quiz 2

1. (a) has a standard normal distribution. Therefore,

has a c² distribution with 1 d.f.

(b) has a standard normal distribution, i = 1, 2. Therefore,

has a c² distribution with 2 d.f.

(c) According to Equation (5) of Sec. 7.4, it has a t distribution with 1 d.f.

2. (a) (X₁+X₂) is sufficient for q (Example 1 of Sec. 6.7). And ½(X₁+X₂) is MLE for q because

Let , we have as a MLE.

Since ½(X₁+X₂) is both a MLE and sufficient for q, ½(X₁+X₂) (or equivalently, (X₁+X₂)) is minimal sufficient.

(b) By the invariance property of MLE, the MLE of qe^-q is

(c) Let . It is unbiased because

(d) The UMVUE . Let Y=X₁+X₂, we have

=

Therefore is the UMVUE for