Download Statistics Module, Week 1 TP(a)

Statistics Module, Week 6 TP(a)
1. It is not known what proportion p of the purchases of a certain brand of muesli are made
by women and what proportion are made by men. In a random sample of 70 purchases of
this muesli, it was found that 58 were made by women and 12 were made by men.
a. Find the MLE of p.
b. Suppose also that it is known that 1/2  p  2/3. Now what is the MLE?
2. This question concerns maximum likelihood estimation of base content of DNA.
Consider the RV X = number of purines in a block of 500 base pairs (bp). Suppose that
X is distributed according to a Poisson distribution with parameter *500, where  is the
purine rate per bp.
a. What are the purines? pyrimidines? (I know, this isn’t a statistics question .)
b. What is E(X)? What is SD(X)?
c. Suppose in a particular data set it was found that in 10 independent blocks of 500
bp, there were the following purine counts: 273, 300, 257, 282, 289, 294, 292,
310, 286, 324. What is the MLE of ?
d. For the same counts as in part (c), what is the MLE of SD(X)?
3. Suppose the RV X comes from a normal distribution with unknown mean and SD. You
have a random sample of size 30 from this distribution; the sample mean is 5.2 with a
sample SD of 2.8. Find the MLE of v = P(X > 2).
4. If gene frequencies are in equilibrium, the genotypes AA, Aa, and aa occur with
probabilities (1-)2, 2 (1-), and 2, respectively. The following data were published on
haptoglobin type in a sample of 190 people:
Haptoglobin Type
Hp1-1
Hp1-2
Hp2-2
10
68
112
a. Find the MLE of .
b. Find the asymptotic variance of the MLE.
[NOTE: You will probably need to consult your notes from this morning.]
5. Consider the two estimators: s12 = (Xi – mean(X))2/ (n-1) and s22 = (Xi – mean(X))2 / n.
a. Which is unbiased?
b. Which has the smaller MSE?