HW3 442Solutions

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Homework 3 Solutions
1. [§6-4] Let X1 , X2 , . . . , X8 be i.i.d. normal random variables with mean µ and standard deviation σ. Define
X −µ
,
T =√
S 2 /n where X is the sample mean and S 2 is the sample variance.
(a) Find τ1 such that P(|T | < τ1 ) = .9; and
(b) find τ2 such that P(T > τ2 ) = .05.
(a) First, we notice that T follows a t7 distribution. Since the t distribution is symmetric about 0, we have
0.9 = P(|T | < τ1 ) = P(T < τ1 ) − P(T < −τ1 ) = 2P(T < τ1 ) − 1.
Thus P(T < τ1 ) = 0.95 and τ1 = 1.895.
(b) Since P(T > τ2 ) = .05, we have P(T < τ2 ) = 0.95. Thus τ2 = 1.895.

2. [§8-4] Suppose that X is a discrete random variable with
2
θ,
3
2
P(X = 2) = (1 − θ),
3
P(X = 0) =

1 θ, 3
1
P(X = 3) = (1 − θ).
3
P(X = 1) =

where 0 ≤ θ ≤ 1 is a parameter. The following 10 independent observations were taken from such a distribution: (3, 0, 2, 1, 3, 2, 1, 0, 2, 1).
(a) Find the method of moments estimate of θ.
(b) What is the maximum likelihood estimate of θ?

(a) In general, let X1 , X2 , . . . , Xn be a random sample drawn from this distribution. Since
2
1
2
1
7
µ1 = E[X] = 0 · θ + 1 · θ + 2 · (1 − θ) + 3 · (1 − θ) = − 2θ.
3
3
3
3
3
we have θ= and the MME for θ is

7 1
− µ1 ,
6 2

7 1 θˆ = − X,
6 2

1∑
Xi . In this case, we have X = 3/2 and n i=1 n where X =

7 1 3
5
θˆ = − · =
.
6 2 2
12
(b) The likelihood function of the sample (3, 0, 2, 1, 3, 2, 1, 0, 2, 1) is lik(θ) = f (3, 0, 2, 1, 3, 2, 1, 0, 2, 1|θ)
( )2 ( )3 (
)3 (
)2
2
1
2
1
= θ · θ ·
(1 − θ) ·
(1 − θ)
3
3
3
3
( )5 ( )5
2
1
5
= θ5 (1 − θ) .
3
3
Since
l(θ) = log lik(θ) = 5 log

1
2
+ 5 log + 5 log θ + 5 log(1 − θ),
3
3

5
5
−
,
θ 1−θ
5
5 l′′ (θ) = − 2 −
< 0, θ (1 − θ)2 l′ (θ) =

we have

( )
1
l
= max l(θ).
0≤θ≤1
2

1
Thus in this case, the MLE is θ˜ = .
2

2

3. [§8-7] Suppose that X follows a geometric distribution,
P(X = k) = p(1 − p)k−1 and assume an i.i.d. sample of size n.
(a) Find the method of moments estimate of p.
(b) Find the mle of p.
(The moments of geometric distribution can be