STAT 100B HW 8 Due Friday
Problem 1:
Suppose
Y
∼
Bernoulli(
λ
). Suppose [
X

Y
= 0]
∼
N(
μ
0
, σ
2
), and [
X

Y
= 1]
∼
N(
μ
1
, σ
2
).
(1) Find the marginal density of
X
. Explain your result intuitively.
(2) Calculate Pr(
Y
= 1

X
=
x
). Explain your result intuitively.
(3) Show that log[Pr(
Y
= 1

X
=
x
)
/
Pr(
Y
= 0

X
=
x
)] =
β
0
+
β
1
x
, where
β
0
and
β
1
can be
calculated from
λ
,
μ
1
,
μ
0
,
σ
2
.
Problem 2:
Suppose
Y
i
∼
Bernoulli(
λ
) independently, for
i
= 1
, ..., n
. Suppose [
X
i

Y
i
= 0]
∼
N(
μ
0
, σ
2
), and [
X
i

Y
i
= 1]
∼
N(
μ
1
, σ
2
). Find the maximum likelihood estimates of
λ
,
μ
1
,
μ
0
,
σ
2
.
Problem 3:
In the following data set, the first column records
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 Spring '11
 Wu
 Bernoulli, Maximum likelihood, Yi, Maximum likelihood estimates

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