312-midterm2sol - Stat312: Midterm I Solution Moo K. Chung,...

Stat312: Midterm I Solution
Moo K. Chung, Yulin Zhang
[email protected], [email protected]
March 5, 2003
1. Let X
1
, · · · , X
n
be a random sample from Bernoulli
distribution with parameter p .
(a) What is ( S
2
/p
2
) ? S
2
is the sample variance. Ex-
plain your results (10 points).
(b) Find an unbiased estimator of p 2 . Explain your
results (5 points).
2. Let X
1
, X
2
be a random sample from N (0 , 1 /θ ) . Note
that the sample size is 2 and the density function for X
i
is f ( x
i
) =
√
θ
√
2 π
exp( - θx 2
i
/ 2) .
(a) Obtain an estimator of θ using the method of mo-
ments. Explain your results (5 points).
(b) Find the likelihood function and use it to obtain the