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.
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- Fall '07
- Wu
- Probability, Trigraph, NAA, xi −x
-
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