Math 304: Homework 2
Due: February 9
1. (WMS 8.8) Suppose that Y1, Y2, Y3denote a random sample from an exponential distribution
with density function
f(x) = 1
θe−x/θ for x > 0.
Consider the following five estimators of θ:
ˆ
θ1=Y1,ˆ
θ2=Y1+Y2
2,ˆ
θ3=Y1+ 2Y2
3,ˆ
θ4= min{Y1, Y2, Y3},ˆ
θ5=¯
Y
(a) Which of these estimators are unbiased?
(b) Among the unbiased estimators, which has the smallest variance?
2. (WMS: 9.89) It is known that the probability pof tossing heads on an unbalanced coin is either
1/4 or 3/4. The coin is tossed twice and a value for Y, the number of heads, is observed. For
each possible value of Y, which of the two values for p(1/4 or 3/4) maximizes the probability
that Y=y? Depending on the value of yactually ovserved, what is the MLE of p?
3. Lavine, Ch2 #11
4. Lavine, Ch2 #12
5. Lavine, Ch2 #13
6. HMC 6.1.2: Let X1,...,Xnbe a random sample from a Gam(α= 3, β =θ) distribution,
0< θ < ∞. Determine the mle of θ.
7. HMC 6.1.7: Let the table
x0 1 2 3 4 5
frequency 6 10 14 13 6 1
represent a summary of a sample of size 50 from a binomial distribution having n= 5. Find
the mle of P(X≥3).
8. Use R to obtain a sample of size 20 from a Poi(3). Obtain a graph of the likelihood function
and the log-likelihood function. Locate the MLE on the graphs.
1