STAT244  HW5 solutions  Due 2/16/2017
Total:
100
pts.
Notation
:
1.
f
✓
(
t
)
denotes the (probability) density (function) of random variable
✓
at
t
,
F
✓
(
t
)
 the (cumulative) distribution (function);
2.
Φ
denotes the distribution of
N
(0
,
1)
;
3. If
n
2
N
, i.e.,
n
is an integer,
[
n
]
denotes
1
,
2
, . . . , n

1
, n
(sometimes denoted
as
1
, n
);
4.
l
(
✓

x
)
will denote the loglikelihood for parameter
✓
and data
x
;
5.
ˆ
✓
MLE
stands for the MLE of
✓
;
6.
1
A
(
x
)
denotes a function that is
1
whenever
x
2
A
, and
0
otherwise.
Problem 1 [15 pts]
Suppose
X
has a Beta
(
a, b
)
density.
1.
[9
pts]
Use
the
normal
approximation
to
find
all
three
of
P

X

.
6

< .
001
,
P

X

.
6

< .
01
and
P

X

.
6

< .
1
when
X
is (a) Beta
(3
,
2)
, (b) Beta
(30
,
20)
, (c) Beta
(300
,
200)
. Present the results
in a
3
⇥
3
table.
Solution
.
X
⇠
Beta
(
a, b
)
)
μ
,
E
[
X
] =
a
a
+
b
;
σ
2
,
Var
[
X
] =
ab
(
a
+
b
)
2
(
a
+
b
+ 1)
For large
a
and
b
,
X
is also approximately normal,
Y
⇠
N
(
μ,
σ
2
)
. Given standard
normal
Z
⇠
N
(0
,
1)
with distribution
Φ
, for any
x
2
R
,
P

Y

μ

< x
=
P
σ

Z

< x
=
P

Z

< x/
σ
=
Φ
(
x/
σ
)

Φ
(

x/
σ
)
[3 pts]
Using this result, we can construct a
3
⇥
3
table [6 pts] with the desired probabilities
for
P

X

.
6

< x
:
x
=
.
001
x
=
.
01
x
=
.
1
a
= 3
, b
= 2
.
004
.
0399
.
3829
a
= 30
, b
= 20
.
0116
.
1159
.
8551
a
= 300
, b
= 200
.
0364
.
3523
1
2. [6 pts] For case (a) only, find the same three probabilities
exactly directly
from the beta density by integration.
Solution
. For
a
= 3
, b
= 2
the density is
f
(
x
) =
Γ
(5)
Γ
(3)
Γ
(2)
x
2
(1

x
) = 12(
x
2

x
3
)
Let
x

=
.
6

x
and
x
+
=
.
6 +
x
, with
x
as in the previous item. We can compute
the exact probabilities by computing the integral:
P

X

.
6

< x
= 12
Z
x
+
x

(
x
2

x
3
)
dx
= 4
x
3
+

3
x
4
+

4
x
3

+ 3
x
4

[3 pts]
We get results [3 pts] that closely resemble the ones in the previous table
x
=
.
001
x
=
.
01
x
=
.
1
a
= 3
, b
= 2
.
0035
.
0346
.
3392
Problem 2 [13 pts]
Suppose there are
n
voters and
n
is even. Suppose voters
vote independently of one another, each with probability
p
of voting for candidate
A
and probability
1

p
of voting for candidate
B
. Let
X
be the total vote for
A
.
1. [2 pts] What is the probaiblity of a tie if
p
=
.
5
and
n
= 20
?
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 Fall '08
 DRTON
 Probability

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