Jointly Discrete Random Variables
• Automobile engines and transmissions are produced
on assembly lines, and are inspected for defects after
they come off their assembly lines. Those with defects
are repaired.
• Let X represent the number of engines, and Y
represent the number of transmissions that require
repairs in a one-hour time interval. Total number of
repairs is X+Y.
• The possible values of X are 0, 1, 2, 3, and the
possible values of Y are 0, 1, 2, 3. So, both X and Y
are discrete and together determine the total number
of repairs needed in a given one-hour time interval.
• We say that X and Y are jointly discrete.
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
1
Joint probability mass function
• There are 16 possible values for the ordered pair:
(0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (1,3),
(2,0), (2,1), (2,2), (2,3), (3,0), (3,1), (3,2), (3,3).
• The probability of each of the ordered pair occurring is given by
the joint probability mass function:
𝑝𝑝(𝑥𝑥, 𝑦𝑦) = 𝑃𝑃 𝑋𝑋 = 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 𝑌𝑌 = 𝑦𝑦
y
0
1
2
3
0
0.13
0.10
0.07
0.03
1
0.12
0.16
0.08
0.04
2
0.02
0.06
0.08
0.04
3
0.01
0.02
0.02
0.02
x
• For example 𝑝𝑝 0, 0 = 0.13, 𝑝𝑝 3, 3 = 0.02
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
2
Joint probability mass function
• Let X be the number of engines and Y be the number of
transmissions that require repairs in a one-hour time interval.
• The possible values of both X are Y are 0, 1, 2, 3. Therefore,
there are 16 possible values for the ordered pair:
(0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (1,3),
(2,0), (2,1), (2,2), (2,3), (3,0), (3,1), (3,2), (3,3).
• The joint probability mass function 𝑝𝑝 𝑥𝑥, 𝑦𝑦 = 𝑃𝑃 (𝑋𝑋 = 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 𝑌𝑌 = 𝑦𝑦)
y
Dr. Sumon
x
0
1
2
3
0
0.13
0.10
0.07
0.03
1
0.12
0.16
0.08
0.04
2
0.02
0.06
0.08
0.04
3
0.01
0.02
0.02
0.02
Section 2.6 Jointly Distributed Random Variables
3
Marginal probability mass function
• The marginal probability mass function of X
𝑝𝑝𝑋𝑋 𝑥𝑥 = 𝑃𝑃 𝑋𝑋 = 𝑥𝑥 = � 𝑝𝑝(𝑥𝑥, 𝑦𝑦)
𝑦𝑦
y
𝑝𝑝𝑋𝑋 𝑥𝑥
𝑝𝑝𝑋𝑋 𝑥𝑥
x
0
1
2
3
0
0.13
0.10
0.07
0.03
0.33
0
0.33
1
0.12
0.16
0.08
0.04
0.40
1
0.40
2
0.02
0.06
0.08
0.04
0.20
2
0.20
3
0.01
0.02
0.02
0.02
0.07
4
0.07
x
𝑝𝑝𝑋𝑋 0 = 𝑃𝑃 𝑋𝑋 = 0 = � 𝑝𝑝 0, 𝑦𝑦 = 𝑝𝑝 0,0 + 𝑝𝑝 0,1 + 𝑝𝑝 0,2 + 𝑝𝑝(0,3)
𝑦𝑦
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
4
Marginal probability mass function
• The marginal probability mass function of Y
𝑝𝑝𝑌𝑌 𝑦𝑦 = 𝑃𝑃 𝑌𝑌 = 𝑦𝑦 = � 𝑝𝑝(𝑥𝑥, 𝑦𝑦)
y
𝑥𝑥
x
0
1
2
3
0
0.13
0.10
0.07
0.03
1
0.12
0.16
0.08
0.04
2
0.02
0.06
0.08
0.04
3
0.01
0.02
0.02
0.02
𝑝𝑝𝑌𝑌 𝑦𝑦
𝟎𝟎. 𝟐𝟐𝟐𝟐
0.34
0.25
0.13
y
𝑝𝑝𝑌𝑌 𝑦𝑦
Dr. Sumon
0
1
2
3
0.28
0.34
0.25
0.13
Section 2.6 Jointly Distributed Random Variables
5
Joint probability mass function
• The joint pmf satisfy the following conditions:
y
x
0
1
2
3
0
1
0.13
0.12
0.10
0.16
0.07
0.08
0.03
0.04
2
0.02
0.06
0.08
0.04
3
0.01
0.02
0.02
0.02
� � 𝑝𝑝 𝑥𝑥, 𝑦𝑦 = 1
𝑥𝑥
𝑦𝑦
� � 𝑝𝑝 𝑥𝑥, 𝑦𝑦 = � 𝑝𝑝 𝑥𝑥, 0 + 𝑝𝑝 𝑥𝑥, 1 + 𝑝𝑝 𝑥𝑥, 2 + 𝑝𝑝 𝑥𝑥, 3
𝑥𝑥
𝑦𝑦
𝑥𝑥
� 𝑝𝑝 𝑥𝑥, 0 = 𝑝𝑝 0,0 + 𝑝𝑝 1,0 + 𝑝𝑝 2,0) + 𝑝𝑝(3,0 = 0.28
𝑥𝑥
� 𝑝𝑝 𝑥𝑥, 1 = 𝑝𝑝 0,1 + 𝑝𝑝 1,1 + 𝑝𝑝 2,1) + 𝑝𝑝(3,1 = 0.34, 𝑒𝑒𝑒𝑒𝑒𝑒.
𝑥𝑥
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
6
Joint probability mass function
• If X and Y are jointly discrete
• The joint pmf of X and Y is the following function:
𝑝𝑝 𝑥𝑥, 𝑦𝑦 = 𝑃𝑃 (𝑋𝑋 = 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 𝑌𝑌 = 𝑦𝑦)
• The marginal pmf of X and Y can be obtained from the
joint pmf as follows:
𝑝𝑝𝑋𝑋 𝑥𝑥 = 𝑃𝑃 𝑋𝑋 = 𝑥𝑥 = � 𝑝𝑝 𝑥𝑥, 𝑦𝑦
𝑦𝑦
𝑝𝑝𝑌𝑌 𝑦𝑦 = 𝑃𝑃 𝑌𝑌 = 𝑦𝑦 = � 𝑝𝑝 𝑥𝑥, 𝑦𝑦
• The joint pmf must satisfy the condition that
𝑥𝑥
� � 𝑝𝑝 𝑥𝑥, 𝑦𝑦 = 1
𝑥𝑥
Dr. Sumon
𝑦𝑦
Section 2.6 Jointly Distributed Random Variables
7
Jointly continuous random variables
• The height H and radius R of a randomly selected cylindrical
can be treated as jointly continuous random variables.
R
H
𝑉𝑉 = 𝜋𝜋𝑅𝑅2 𝐻𝐻
• The lifetimes, in months, of two components in a system,
denoted by X and Y. Then, X and Y can be treated as jointly
continuous.
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
8
Joint probability density function
• If X and Y are jointly continuous random variables, their
joint probability density function, f(x,y), has three properties.
• 𝑓𝑓 𝑥𝑥, 𝑦𝑦 ≥ 0 for all x and y
f( x, y)
y
(x,y)
x
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
9
Joint probability density function
• If X and Y are jointly continuous random variables, with
joint probability density function, f(x,y), and a < b, c < d,
then, the probability that 𝑎𝑎 ≤ 𝑋𝑋 ≤ 𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 ≤ 𝑌𝑌 ≤ 𝑑𝑑 is
𝑏𝑏
𝑑𝑑
𝑃𝑃 𝑎𝑎 ≤ 𝑋𝑋 ≤ 𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 ≤ 𝑌𝑌 ≤ 𝑑𝑑 = � � 𝑓𝑓 𝑥𝑥, 𝑦𝑦 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝑎𝑎
d
y
𝑐𝑐
c
a
x
b
• The joint pdf satisfy the following condition:
Dr. Sumon
�
∞
∞
� 𝑓𝑓 𝑥𝑥, 𝑦𝑦 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 1
−∞ −∞
Section 2.6 Jointly Distributed Random Variables
10
Marginal probability density function
• If X and Y are jointly continuous random
variables, with joint probability density function,
f(x,y), then the marginal probability density
function of X and Y are given, respectively, by
𝑓𝑓𝑋𝑋 𝑥𝑥 = �
𝑦𝑦=∞
𝑓𝑓 𝑥𝑥, 𝑦𝑦 𝑑𝑑𝑑𝑑
𝑦𝑦=−∞
𝑓𝑓𝑌𝑌 𝑦𝑦 = �
𝑥𝑥=∞
𝑓𝑓 𝑥𝑥, 𝑦𝑦 𝑑𝑑𝑑𝑑
𝑥𝑥=−∞
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
11
Example 2.54 and 2.55
• Assume that for certain type of washer, both the
thickness and the hole diameter vary from item to
item. Let X denote the thickness in millimeters
and let Y denote the hole diameter in millimeters,
for a randomly chosen washer.
Assume that the joint density function
1
𝑓𝑓 𝑥𝑥, 𝑦𝑦 = �6 𝑥𝑥 + 𝑦𝑦 𝑖𝑖𝑖𝑖 1 ≤ 𝑥𝑥 ≤ 2 𝑎𝑎𝑎𝑎𝑎𝑎 4 ≤ 𝑦𝑦 ≤ 5
0
𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
12
Example 2.54
• Find the probability that a randomly chosen
washer has a thickness between 1.0 and 1.5 mm,
and a hole diameter between 4.5 and 5 mm.
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
13
SOLUTION
• Find 𝑃𝑃 1 ≤ 𝑋𝑋 ≤ 1.5 𝑎𝑎𝑎𝑎𝑎𝑎 4.5 ≤ 𝑌𝑌 ≤ 5
Recall 𝑃𝑃 𝑎𝑎 ≤ 𝑋𝑋 ≤ 𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 ≤ 𝑌𝑌 ≤ 𝑑𝑑
𝑏𝑏
= � � 𝑓𝑓 𝑥𝑥, 𝑦𝑦 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝑎𝑎
𝑃𝑃 1 ≤ 𝑋𝑋 ≤ 1.5 𝑎𝑎𝑎𝑎𝑎𝑎 4.5 ≤ 𝑌𝑌 ≤ 5 = �
Given, 𝑓𝑓 𝑥𝑥, 𝑦𝑦 =
5
1
�6
0
Dr. Sumon
𝑐𝑐
5
� 𝑓𝑓(𝑥𝑥, 𝑦𝑦)𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
4.5
𝑥𝑥 + 𝑦𝑦 𝑖𝑖𝑖𝑖 1 ≤ 𝑥𝑥 ≤ 2 𝑎𝑎𝑎𝑎𝑎𝑎 4 ≤ 𝑦𝑦 ≤ 5
𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
𝑃𝑃 1 ≤ 𝑋𝑋 ≤ 1.5 𝑎𝑎𝑎𝑎𝑎𝑎 4.5 ≤ 𝑌𝑌 ≤ 5
4
1.5
x
1.5
1
y 4.5
1
𝑑𝑑
2
=�
1.5
1
5
1
�
𝑥𝑥 + 𝑦𝑦 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
6
4.5
Section 2.6 Jointly Distributed Random Variables
14
SOLUTION
𝑃𝑃 1 ≤ 𝑋𝑋 ≤ 1.5 𝑎𝑎𝑎𝑎𝑎𝑎 4.5 ≤ 𝑌𝑌 ≤ 5
5
=�
1.5
1
y 4.5
4
1
1.5
x
2
=�
1.5
1
=�
1.5
1
5
1
�
𝑥𝑥 + 𝑦𝑦 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
6
4.5
𝑥𝑥𝑥𝑥 𝑦𝑦 2
+
6 12
𝑦𝑦=4.5
𝑥𝑥
19
+
𝑑𝑑𝑑𝑑
12 48
𝑥𝑥 2 19𝑥𝑥
+
=
24 48
Dr. Sumon
𝑦𝑦=5
𝑥𝑥=1.5
𝑥𝑥=1
𝑑𝑑𝑑𝑑
1
=
4
Section 2.6 Jointly Distributed Random Variables
15
Example 2.55
• Find the marginal probability density function of
the thickness X of a washer and the marginal
probability density function of the hole diameter Y
of a washer.
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
16
SOLUTION
• Find 𝑓𝑓𝑋𝑋 𝑥𝑥
Given, 𝑓𝑓 𝑥𝑥, 𝑦𝑦 =
𝑓𝑓𝑋𝑋 𝑥𝑥 = �
𝑦𝑦=∞
𝑦𝑦=−∞
=�
𝑦𝑦=5
𝑦𝑦=4
0
𝑥𝑥 + 𝑦𝑦 𝑖𝑖𝑖𝑖 1 ≤ 𝑥𝑥 ≤ 2 𝑎𝑎𝑎𝑎𝑎𝑎 4 ≤ 𝑦𝑦 ≤ 5
𝑓𝑓 𝑥𝑥, 𝑦𝑦 𝑑𝑑𝑦𝑦
1
𝑥𝑥 + 𝑦𝑦 𝑑𝑑𝑦𝑦 𝑓𝑓𝑓𝑓𝑓𝑓 1 ≤ 𝑥𝑥 ≤ 2
6
1
𝑦𝑦 2
=
𝑥𝑥𝑥𝑥 +
2
6
Dr. Sumon
1
�6
5
y 4.5
4
𝑦𝑦=5
𝑦𝑦=4
1
9
𝑓𝑓𝑓𝑓𝑓𝑓 1 ≤ 𝑥𝑥 ≤ 2
=
𝑥𝑥 +
6
2
𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
Section 2.6 Jointly Distributed Random Variables
1
1.5
x
2
17
SOLUTION
• Find 𝑓𝑓𝑌𝑌 𝑦𝑦
Given, 𝑓𝑓 𝑥𝑥, 𝑦𝑦 =
𝑓𝑓𝑌𝑌 𝑦𝑦 = �
𝑥𝑥=∞
𝑥𝑥=−∞
=�
𝑥𝑥=2
𝑥𝑥=1
0
𝑥𝑥 + 𝑦𝑦 𝑖𝑖𝑖𝑖 1 ≤ 𝑥𝑥 ≤ 2 𝑎𝑎𝑎𝑎𝑎𝑎 4 ≤ 𝑦𝑦 ≤ 5
𝑓𝑓 𝑥𝑥, 𝑦𝑦 𝑑𝑑𝑥𝑥
1
𝑥𝑥 + 𝑦𝑦 𝑑𝑑𝑥𝑥 𝑓𝑓𝑓𝑓𝑓𝑓 4 ≤ 𝑦𝑦 ≤ 5
6
1 𝑥𝑥 2
=
+ 𝑥𝑥𝑥𝑥
6 2
Dr. Sumon
1
�6
5
y 4.5
4
𝑥𝑥=2
𝑥𝑥=1
1
3
𝑓𝑓𝑓𝑓𝑓𝑓 4 ≤ 𝑦𝑦 ≤ 5
=
𝑦𝑦 +
6
2
𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
Section 2.6 Jointly Distributed Random Variables
1
1.5
x
2
18
Conditional distributions
• Let X be the number of engines and Y be the number of
transmissions that require repairs in a one-hour time interval.
• The joint probability mass function 𝑝𝑝 𝑥𝑥, 𝑦𝑦 = 𝑃𝑃 (𝑋𝑋 = 𝑥𝑥, 𝑌𝑌 = 𝑦𝑦)
and the marginal probability mass functions of X and Y, are
given below.
y
𝑝𝑝𝑋𝑋 𝑥𝑥
x
0
1
2
3
x
0
0.13
0.10
0.07
0.03
0
0.33
1
0.12
0.16
0.08
0.04
1
0.40
2
0.02
0.06
0.08
0.04
2
0.20
3
0.01
0.02
0.02
0.02
3
0.07
y
0
1
2
3
𝑝𝑝𝑌𝑌 𝑦𝑦
0.28
0.34
0.25
0.13
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
19
Conditional distributions
• If we are interested in the probability 𝑃𝑃 𝑋𝑋 = 1 for a given
one-hour time interval, we can use the marginal pmf of X to
determine this probability.
𝑃𝑃 𝑋𝑋 = 1 = 𝑝𝑝𝑋𝑋 1 = 0.40
• Now if we learn Y = 2 in that time interval. This knowledge
changes the probability 𝑃𝑃 𝑋𝑋 = 1 . We need to compute the
conditional probability, 𝑃𝑃 𝑋𝑋 = 1|𝑌𝑌 = 2 .
y
Dr. Sumon
x
0
1
2
3
0
0.13 0.10 0.07 0.03
1
0.12 0.16 0.08 0.04
2
0.02 0.06 0.08 0.04
3
0.01 0.02 0.02 0.02
Section 2.6 Jointly Distributed Random Variables
20
Conditional distributions
• Recall that for two events A and B,
𝑃𝑃 𝐴𝐴 ∩ 𝐵𝐵
𝑃𝑃 𝐴𝐴|𝐵𝐵 =
𝑃𝑃 𝐵𝐵
𝑃𝑃 𝑋𝑋 = 1, 𝑌𝑌 = 2
• Similarly, 𝑃𝑃 𝑋𝑋 = 1|𝑌𝑌 = 2 =
𝑃𝑃 𝑌𝑌 = 2
y
𝑝𝑝(1,2)
=
0
1
2
3
x
𝑝𝑝𝑌𝑌 (2)
0
0.07
1
0.08
2
0.08
3
0.02
y
𝑝𝑝𝑌𝑌 𝑦𝑦
Dr. Sumon
0
1
2
3
0.08
= 0.32
=
0.25
0.25
Section 2.6 Jointly Distributed Random Variables
21
Conditional distributions (discrete)
• Let X and Y be jointly discrete random variables, with
joint probability mass function 𝑝𝑝(𝑥𝑥, 𝑦𝑦).
• The conditional probability mass function of Y given 𝑋𝑋 = 𝑥𝑥:
𝑝𝑝𝑌𝑌|𝑋𝑋
𝑝𝑝 𝑥𝑥, 𝑦𝑦
𝑦𝑦 𝑥𝑥 = 𝑃𝑃 𝑌𝑌 = 𝑦𝑦 𝑋𝑋 = 𝑥𝑥 =
,
𝑝𝑝𝑋𝑋 𝑥𝑥
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑡𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑋𝑋 (𝑥𝑥) > 0
• The conditional probability mass function of X given 𝑌𝑌 = 𝑦𝑦:
𝑝𝑝(𝑥𝑥, 𝑦𝑦)
,
𝑝𝑝𝑋𝑋|𝑌𝑌 𝑥𝑥 𝑦𝑦 = 𝑃𝑃 𝑋𝑋 = 𝑥𝑥 𝑌𝑌 = 𝑦𝑦 =
𝑝𝑝𝑌𝑌 (𝑦𝑦)
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑡𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑌𝑌 (𝑦𝑦)>0
• We use conditional pmf to calculate conditional
probability (and conditional mean, etc.).
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
22
Conditional distributions (continuous)
• Let X and Y be jointly continuous random variables, with
joint probability density function 𝑓𝑓(𝑥𝑥, 𝑦𝑦).
The conditional probability density function of Y given 𝑋𝑋 = 𝑥𝑥:
𝑓𝑓(𝑥𝑥, 𝑦𝑦)
𝑓𝑓𝑌𝑌|𝑋𝑋 (𝑦𝑦|𝑥𝑥) =
, provided the marginal pdf 𝑓𝑓𝑋𝑋 𝑥𝑥 > 0.
𝑓𝑓𝑋𝑋 (𝑥𝑥)
• The conditional probability density function of X given 𝑌𝑌 = 𝑦𝑦:
𝑓𝑓(𝑥𝑥, 𝑦𝑦)
𝑓𝑓𝑋𝑋|𝑌𝑌 (𝑥𝑥|𝑦𝑦) =
, provided the marginal pdf 𝑓𝑓𝑌𝑌 𝑦𝑦 > 0.
𝑓𝑓𝑌𝑌 (𝑦𝑦)
• We have
𝑏𝑏
𝑃𝑃 𝑎𝑎 ≤ 𝑋𝑋 ≤ 𝑏𝑏 | 𝑌𝑌 = 𝑦𝑦 = � 𝑓𝑓𝑋𝑋|𝑌𝑌 (𝑥𝑥|𝑦𝑦) 𝑑𝑑𝑑𝑑
𝑎𝑎
𝑑𝑑
Dr. Sumon
𝑃𝑃 𝑐𝑐 ≤ 𝑌𝑌 ≤ 𝑑𝑑 | 𝑋𝑋 = 𝑥𝑥 = � 𝑓𝑓𝑌𝑌|𝑋𝑋 (𝑦𝑦|𝑥𝑥) 𝑑𝑑𝑦𝑦
𝑐𝑐
Section 2.6 Jointly Distributed Random Variables
23
Means of functions of single random variables
• Let X be a random variable, and ℎ 𝑋𝑋 be a function of X,
e.g., ℎ 𝑋𝑋 = 20𝑋𝑋, ℎ 𝑋𝑋 = 𝑋𝑋 2 − 5, 𝑒𝑒𝑒𝑒𝑒𝑒.
• If X is discrete with probability mass function 𝑝𝑝 𝑥𝑥 , the
mean of ℎ 𝑋𝑋 is given by
𝐸𝐸 ℎ 𝑋𝑋
= 𝜇𝜇ℎ(𝑋𝑋) = � ℎ 𝑥𝑥 𝑝𝑝(𝑥𝑥)
𝑥𝑥
The summation is over all possible values of X.
• If X is continuous with probability density function 𝑓𝑓(𝑥𝑥),
the mean of ℎ 𝑋𝑋 is given by
𝜇𝜇ℎ(𝑋𝑋) = �
∞
−∞
Dr. Sumon
ℎ 𝑥𝑥 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑
Section 2.6 Jointly Distributed Random Variables
24
Means of functions of two random variables
• If X and Y are jointly discrete random variables with joint
probability mass function p(x,y),
𝜇𝜇ℎ(𝑋𝑋,𝑌𝑌) = � � ℎ 𝑥𝑥, 𝑦𝑦 𝑝𝑝(𝑥𝑥, 𝑦𝑦)
𝑥𝑥
𝑦𝑦
where the sum is taken over all possible values of X and Y
• If X and Y are jointly continuous random variables with
joint probability density function f(x,y),
𝜇𝜇ℎ(𝑋𝑋,𝑌𝑌) = �
∞
�
∞
−∞ −∞
Dr. Sumon
ℎ 𝑥𝑥, 𝑦𝑦 𝑓𝑓 𝑥𝑥, 𝑦𝑦 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
Section 2.6 Jointly Distributed Random Variables
25
Conditional Means
• If X and Y are jointly discrete random variables with joint probability
mass function p(x,y), the conditional expectation of Y given X= x:
𝐸𝐸(𝑌𝑌|𝑋𝑋 = 𝑥𝑥) = � 𝑦𝑦 𝑝𝑝𝑌𝑌|𝑋𝑋 (𝑦𝑦|𝑥𝑥)
𝑦𝑦
𝐸𝐸(𝑌𝑌) = � 𝑦𝑦 𝑝𝑝𝑌𝑌 (𝑦𝑦)
𝑦𝑦
• If X and Y are jointly continuous random variables with joint probability
density function f(x,y), the conditional expectation of Y given X=x:
𝐸𝐸(𝑌𝑌|𝑋𝑋 = 𝑥𝑥) = �
∞
−∞
∞
Dr. Sumon
𝐸𝐸(𝑌𝑌) = �
−∞
𝑦𝑦 𝑓𝑓𝑌𝑌|𝑋𝑋 𝑦𝑦|𝑥𝑥 𝑑𝑑𝑑𝑑
𝑦𝑦 𝑓𝑓𝑌𝑌 𝑦𝑦 𝑑𝑑𝑑𝑑
Section 2.6 Jointly Distributed Random Variables
26
Independent random variables
• If X and Y are jointly discrete, they are independent if the
joint probability mass function is equal to the product of
the marginals:
𝑝𝑝(𝑥𝑥, 𝑦𝑦) = 𝑝𝑝𝑋𝑋 (𝑥𝑥) 𝑝𝑝𝑌𝑌 (𝑦𝑦)
• If X and Y are jointly continuous, they are independent if
the joint probability density function is equal to the
product of the marginals:
𝑓𝑓(𝑥𝑥, 𝑦𝑦) = 𝑓𝑓𝑋𝑋 (𝑥𝑥) 𝑓𝑓𝑌𝑌 (𝑦𝑦)
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
27
Covariance
• The covariance is a measure of strength of linear
relationship between X and Y.
𝐶𝐶𝐶𝐶𝐶𝐶 𝑋𝑋, 𝑌𝑌 = 𝜇𝜇(𝑋𝑋−𝜇𝜇𝑋𝑋 )(𝑌𝑌−𝜇𝜇𝑌𝑌 )
𝑋𝑋 − 𝜇𝜇𝑋𝑋 = Deviation of X from mean of X
(𝑋𝑋 − 𝜇𝜇𝑋𝑋 )(𝑌𝑌 − 𝜇𝜇𝑌𝑌 ) = Product of deviations of X and Y
• Consider a simple random sample:
𝑥𝑥𝑛𝑛 , 𝑦𝑦𝑛𝑛
Y
Both deviations positive,
Product of deviations positive.
Negative product
Positive product
Dr. Sumon
𝑥𝑥1 , 𝑦𝑦1 , 𝑥𝑥2 , 𝑦𝑦2 ,…,
(𝜇𝜇𝑋𝑋 , 𝜇𝜇𝑌𝑌 )
X
Negative product
Section 2.6 Jointly Distributed Random Variables
28
Covariance
E[(𝑋𝑋 − 𝜇𝜇𝑋𝑋 )(𝑌𝑌 − 𝜇𝜇𝑌𝑌 )]
Association of X and Y in (X,Y)
Larger X tends to pair with larger Y and Tends to be positive
smaller X tend to pair with smaller Y
Negative product
Y
Positive product of deviations of X and Y
(𝜇𝜇𝑋𝑋 , 𝜇𝜇𝑌𝑌 )
Positive product
Dr. Sumon
X
Negative product
Section 2.6 Jointly Distributed Random Variables
29
Covariance
Association of X and Y in (X,Y)
Large X tends to pair with small Y and
vice versa.
Negative product
E[(𝑋𝑋 − 𝜇𝜇𝑋𝑋 )(𝑌𝑌 − 𝜇𝜇𝑌𝑌 )]
Tends to be negative
Y
Positive product
(𝜇𝜇𝑋𝑋 , 𝜇𝜇𝑌𝑌 )
Positive product
Dr. Sumon
X
Negative product
Section 2.6 Jointly Distributed Random Variables
30
Covariance
Association of X and Y in (X,Y)
E[(𝑋𝑋 − 𝜇𝜇𝑋𝑋 )(𝑌𝑌 − 𝜇𝜇𝑌𝑌 )]
Little tendency for larger X to pair with Tends to be zero
either larger Y or smaller Y
Y
Positive product
Negative product
(𝜇𝜇𝑋𝑋 , 𝜇𝜇𝑌𝑌 )
Positive product
Dr. Sumon
X
Negative product
Section 2.6 Jointly Distributed Random Variables
31
Correlation
• The population correlation or simply correlation between
X and Y is denoted by 𝜌𝜌𝑋𝑋,𝑌𝑌 and given by
𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌)
𝜌𝜌𝑋𝑋,𝑌𝑌 =
𝜎𝜎𝑋𝑋 𝜎𝜎𝑌𝑌
• The correlation is free of the units. So we can compare
correlation between two pairs of random variables.
• For any two random variables X and Y
−1 ≤ 𝜌𝜌𝑋𝑋𝑋𝑋 ≤ 1
When there is an exact linear dependency, 𝑌𝑌 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏, then
𝜌𝜌𝑋𝑋𝑋𝑋 = 1 if 𝑏𝑏 > 0 and 𝜌𝜌𝑋𝑋𝑋𝑋 = −1 if 𝑏𝑏 < 0.
Dr. Sumon
Section 2.6 Jointly Distributed Random Variables
32
Formulas
𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌) 𝜇𝜇(𝑋𝑋−𝜇𝜇𝑋𝑋 )(𝑌𝑌−𝜇𝜇𝑌𝑌) 𝜇𝜇𝑋𝑋𝑋𝑋 − 𝜇𝜇𝑋𝑋 𝜇𝜇𝑌𝑌
𝜌𝜌𝑋𝑋𝑋𝑋 =
=
=
𝜎𝜎𝑋𝑋 𝜎𝜎𝑌𝑌
𝜎𝜎𝑋𝑋 𝜎𝜎𝑌𝑌
𝜎𝜎𝑋𝑋 𝜎𝜎𝑌𝑌
• If X and Y are jointly discrete,
𝜇𝜇𝑋𝑋𝑋𝑋 = � � 𝑥𝑥𝑥𝑥 𝑝𝑝(𝑥𝑥, 𝑦𝑦)
𝑥𝑥
𝑦𝑦
𝜇𝜇𝑋𝑋 = � 𝑥𝑥 𝑝𝑝𝑋𝑋 (𝑥𝑥)
𝑥𝑥
𝜇𝜇𝑌𝑌 = � 𝑦𝑦 𝑝𝑝𝑌𝑌 (𝑦𝑦)
𝑦𝑦
Dr. Sumon
𝜎𝜎𝑋𝑋2 = 𝜇𝜇𝑋𝑋 2 − 𝜇𝜇𝑋𝑋
2
𝜇𝜇𝑋𝑋 2 = � 𝑥𝑥 2 𝑝𝑝𝑋𝑋 (𝑥𝑥)
𝑥𝑥
𝜎𝜎𝑌𝑌2 = 𝜇𝜇𝑌𝑌 2 − 𝜇𝜇𝑌𝑌
2
𝜇𝜇𝑌𝑌 2 = � 𝑦𝑦 2 𝑝𝑝𝑌𝑌 (𝑦𝑦)
𝑦𝑦
Section 2.6 Jointly Distributed Random Variables
33
Formula
𝜌𝜌𝑋𝑋𝑋𝑋
𝜇𝜇(𝑋𝑋−𝜇𝜇𝑋𝑋 )(𝑌𝑌−𝜇𝜇𝑌𝑌) 𝜇𝜇𝑋𝑋𝑋𝑋 − 𝜇𝜇𝑋𝑋 𝜇𝜇𝑌𝑌
𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌)
=
=
=
𝜎𝜎𝑋𝑋 𝜎𝜎𝑌𝑌
𝜎𝜎𝑋𝑋 𝜎𝜎𝑌𝑌
𝜎𝜎𝑋𝑋 𝜎𝜎𝑌𝑌
• If X and Y are jointly continuous
𝜇𝜇𝑋𝑋𝑋𝑋 = �
𝜇𝜇𝑋𝑋 = �
∞
−∞ −∞
∞
−∞
∞
𝜇𝜇𝑌𝑌 = �
−∞
Dr. Sumon
�
∞
𝑥𝑥𝑥𝑥 𝑓𝑓(𝑥𝑥, 𝑦𝑦)𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝑥𝑥 𝑓𝑓𝑋𝑋 𝑥𝑥 𝑑𝑑𝑥𝑥
𝑦𝑦 𝑓𝑓𝑌𝑌 𝑦𝑦 𝑑𝑑𝑑𝑑
𝜎𝜎𝑋𝑋2 = 𝜇𝜇𝑋𝑋 2 − 𝜇𝜇𝑋𝑋
𝜇𝜇𝑋𝑋 2 = �
∞
𝑥𝑥 2 𝑓𝑓𝑋𝑋 𝑥𝑥 𝑑𝑑𝑥𝑥
−∞
𝜎𝜎𝑌𝑌2 =
∞
𝜇𝜇𝑌𝑌 2 = �
−∞
Section 2.6 Jointly Distributed Random Variables
𝜇𝜇𝑌𝑌 2 − 𝜇𝜇𝑌𝑌
𝑦𝑦 2 𝑓𝑓𝑌𝑌 𝑦𝑦 𝑑𝑑𝑦𝑦
2
2
34