Stats 100A Fall 07 Midterm - STAT 100B Midterm Solution...

STAT 100B Midterm Solution
Problem 1 Suppose X
1
, X
2
, ..., X
n
∼ Exp( λ ). The density function of Exp( λ ) is f ( x ) =
λe
- λx
for x ≥ 0, and f ( x ) = 0 for x < 0.
(1) Suppose we want to estimate λ by solving the estimating equation Pr( X > 1) =
m/n , where m is the number of X
i
that are greater than 1. Please find the estimate
of λ .
(2) Please find the MLE of λ .
1. Pr ( X > 1) =
R
∞
1
λe - λx = - e - λx | ∞
1
= e - λ = m/n
ˆ
λ = log( n/m )
2. MLE
L ( λ ) = λ n e - λ
∑
x
i
l ( λ ) = n log λ - λ
∑
x
i
∂l
∂λ
= n
λ
-
∑
x
i
= 0
ˆ
λ = 1 /
¯
X
Problem 2 (1) Suppose X
1
, X
2
, ..., X
n
∼ N( μ, σ 2 ). Please find the MLE of μ and σ 2 . Note:
the density function of N( μ, σ 2 ) is f ( x ) =
1
√
2 πσ 2
e
-
( x - μ )
2
2 σ
2
.
(2) Suppose X
1
, X
2
, ..., X
n
∼ N( μ
1
, σ 2 ), and Y
1
, Y
2
, ..., Y
m
∼ N( μ
2
, σ 2 ). Please find the
MLE of μ
1
, μ
2
and σ 2 .
1. X
1
, · · · , X
n
∼ N ( μ, σ 2 )
L ( μ, σ 2 ) = (2 πσ 2 )
-
n
2
e
-
∑
( x
i
- μ )
2
2 σ
2
l ( μ, σ
2
) = -
n
2
log 2 πσ
2
-
∑
( x
i
- μ )
2
2 σ 2
∂l
∂μ
=
∑
( x
i
- μ )
σ 2
= 0
∂l
∂σ 2
= -
n
2
1
σ 2
+
∑
( x
i
- μ )
2
2 σ 4
= 0
ˆ μ
MLE
=
¯
X, ˆ σ
2
MLE
=
∑
( x
i
-
¯
X )
2
n
2.