MA 967
Homework 1
W. Yu
Due: Thursday, December 23
1. Exercise 4.1.3
Suppose the number of customers X that enter a store between the hours 9:00 a.m. and
10:00 a.m. follows a Poisson distribution with parameter θ. Suppose a random sample
of the number of customers that enter the store between 9:00 a.m. and 10:00 a.m. for
10 days results in the values
9 7 9 15 10 13 11 7 2 12
(a) Determine the maximum likelihood estimator of θ. Show that it is an unbiased
estimator.
(b) Based on these data, obtain the realization of your estimator in part (a). Explain
the meaning of this estimate in terms of the number of customers.
2. Exercise 4.1.4 (See Page 2)
For Example 4.1.3, verify equations (4.1.4) - (4.1.8).
3. Let X1, · · · , Xn be a random sample. Find the mle of θ for the cases that the population
pdf/pmf is,
Page 1 of 2
MA 967
Homework 1
W. Yu
Due: Thursday, December 23
a) f(x; θ) =
{
θ · xθ−1, 0 6 x 6 1, θ > 0
0, otherwise
b) f(x; θ) =
{
θ−12 · e−(x−θ1)/θ2 , x > θ1, θ2 > 0
0, otherwise
4. Exercise 4.2.4 a), b)
Suppose we assume that X1, X2, ..., Xn is a random sample from a Γ(1, θ) distribution.
(a) Show that the random variable (2/θ)
∑n
i=1Xi has a χ
2−distribution with 2n degrees
of freedom.
(b) Using the random variable in part (a) as pivot random variable, find a (1− α)100%
confidence interval for θ.
5. Exercise 4.2.12
Let Y be B(300, p). If the observed value of Y is y = 75, find an approximate 90%
confidence interval for p.
6. Exercise 4.2.19
Let X1, X2, ..., Xn be a random sample from a gamma distribution with known param-
eter α = 3 and unknown β > 0. Discuss the construction of a confidence interval for
β.
Hint: What is the distribution of 2
∑n
1 Xi/β?
7. Exercise 4.4.5
Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size 4 from the
distribution having pdf f(x) = e−x, 0 < x <∞, zero elsewhere. Find P (Y4 > 3).
8. Exercise 4.4.15 a)
Let Y1 < Y2 denote the order statistics of a random sample of size 2 from N(0, σ
2).
(a) Show that E(Y1) = −σ/
√
(pi).
Hint: Evaluate E(Y1) by using the joint pdf of Y1 and Y2 and first integrating on y1.