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MA 967

Homework 1

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,

a) f (x; θ) =

b) f (x; θ) = _0(θ)1 . e − (北

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) Xi has a χ2 -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 1(n) 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 −北 , 0 < x < o, 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, σ2 ). (a) Show that E(Y1 ) = -σ/′ (π).

Hint: Evaluate E(Y1 ) by using the joint pdf of Y1 and Y2 and ﬁrst integrating on y1 .