Example Class 3 (sln) - The University of Hong Kong...

The University of Hong Kong
Department of Statistics and Actuarial Science
STAT1801 Probability and Statistics: Foundations of Actuarial Science (10-11)
Example Class 3 Solution
1. Suppose random variables 𝑋𝑋 and 𝑌𝑌 have the joint pdf
𝑓𝑓
(
𝑥𝑥 , 𝑦𝑦
) = 3(
𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦
2
)
for 𝑥𝑥 , 𝑦𝑦 ∈ [0,1] .
(a) Find 𝑃𝑃
(
𝑌𝑌 ≥ 2 𝑋𝑋
)
.
𝑃𝑃 ( 𝑌𝑌 ≥ 2 𝑋𝑋 ) = 3 � � ( 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦
2
)
1
2 𝑥𝑥
𝑑𝑑𝑦𝑦
1 2
⁄
0
𝑑𝑑𝑥𝑥
= 3 � �
𝑥𝑥𝑦𝑦 − 𝑥𝑥𝑦𝑦
2
+
𝑦𝑦
3
3
�
𝑦𝑦 =2 𝑥𝑥
𝑦𝑦 =1
1 2 ⁄
0
𝑑𝑑𝑥𝑥
= 3 �
( 𝑥𝑥 − 𝑥𝑥 +
1
3
1 2
⁄
0
− 2 𝑥𝑥
2
+ 4 𝑥𝑥
3
+
8 𝑥𝑥
3
3
) 𝑑𝑑𝑥𝑥
= 3 �
𝑥𝑥
3
−
2 𝑥𝑥
3
3
+ 𝑥𝑥
4
+
8 𝑥𝑥
4
12
�
0
1
2
=
5
16
(b) Find the marginal pdfs of 𝑋𝑋 and 𝑌𝑌 respectively.
𝑓𝑓
𝑋𝑋
( 𝑥𝑥 ) = � 3( 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦
2
)
1
0
𝑑𝑑𝑦𝑦
= 3 �𝑥𝑥𝑦𝑦 − 𝑥𝑥𝑦𝑦
2
+
𝑦𝑦
3
3
�
0
1
= 1
𝑓𝑓
𝑌𝑌
(
𝑦𝑦
) =
� 3( 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦
2 )
1
0
𝑑𝑑𝑥𝑥
= 3 �
𝑥𝑥
2
2
− 𝑥𝑥
2
𝑦𝑦 + 𝑥𝑥𝑦𝑦
2
�
0
1
= 3 𝑦𝑦
2
− 3 𝑦𝑦 +
3
2
(c) Are 𝑋𝑋 and 𝑌𝑌 independent?
𝑓𝑓
𝑋𝑋
( 𝑥𝑥 ) 𝑓𝑓
𝑌𝑌
( 𝑦𝑦 ) ≠ 𝑓𝑓 ( 𝑥𝑥 , 𝑦𝑦 ) ⇒ Not Independent
(d) Find 𝑓𝑓
𝑋𝑋 | 𝑌𝑌
(
𝑥𝑥
|
𝑦𝑦
)
.
𝑓𝑓
𝑋𝑋 | 𝑌𝑌
( 𝑥𝑥 | 𝑦𝑦 ) =
𝑓𝑓
(
𝑥𝑥 , 𝑦𝑦
)
𝑓𝑓
𝑌𝑌
( 𝑦𝑦 )
=
3( 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦
2
)
3 𝑦𝑦 2 − 3 𝑦𝑦 +
3
2

=
𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦
2
𝑦𝑦 2 − 𝑦𝑦 +
1
2
2. Suppose 𝑋𝑋 and 𝑌𝑌 are discrete random variables with a joint pmf as following:
X
-1 0 1
-1 1/6 1/9 1/9
Y 0 1/9 0 c
1 1/18 1/9 1/6
(a) Find the value of c.