STAT 100B HWVI Solution
Problem 1:
Suppose
X
1
, X
2
, ..., X
n
∼
Bernoulli(
p
) independently. Please calculate the maximum
likelihood estimate of
p
.
A: The likelihood is
L
(
p
) =
Q
n
i
=1
p
X
i
(1

p
)
1

X
i
=
p
∑
n
i
=1
X
i
(1

p
)
n

∑
n
i
=1
X
i
. The loglikelihood
is
l
(
p
) = log
L
(
p
) =
∑
n
i
=1
X
i
log
p
+(
n

∑
n
i
=1
X
i
) log(1

p
). By setting
l
0
(
p
) = 0, we get
∑
n
i
=1
X
i
/p

(
n

∑
n
i
=1
X
i
)
/
(1

p
) = 0. So ˆ
p
=
∑
n
i
=1
X
i
/n
.
Problem 2:
Suppose
X
1
, X
2
, ..., X
n
∼
N(
μ, σ
2
) independently.
Please calculate the maximum
likelihood estimate of (
μ, σ
2
).
A: The likelihood is
L
(
μ, σ
2
)
=
n
Y
i
=1
(
1
√
2
πσ
2
exp
{
(
X
i

μ
)
2
2
σ
2
}
)
=
1
(2
πσ
2
)
n/
2
exp
{
1
2
σ
2
n
X
i
=1
(
X
i

μ
)
2
}
.
The loglikelihood is
l
(
μ, σ
2
)
=

n
2
log(2
πσ
2
)

1
2
σ
2
n
X
i
=1
(
X
i

μ
)
2
.
By setting
∂l
(
μ, σ
2
)
/∂μ
= 0, we have
∑
n
i
=1
(
X
i

μ
) = 0, so ˆ
μ
=
¯
X
.
Let
τ
=
σ
2
.
By setting
∂l
(
μ, τ
)
/∂τ
= 0, we have

n/τ
+
∑
n
i
=1
(
X
i

μ
)
2
/τ
2
= 0, so ˆ
σ
2
=
∑
n
i
=1
(
X
i

ˆ
μ
)
2
/n
.
Problem 3:
Suppose
X
1
, X
2
, ..., X
n
∼
Exponential(
λ
) independently. Please calculate the maxi
mum likelihood estimate of
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 Fall '08
 staff
 Bernoulli, Probability

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