Discrete Variance, Uniform Distribution,
Permutations and Combinations
Instructor: Andrew Liu
September 21, 2011
Textbook sections: 3-4, 3-5, 2-1.4
Review Variance: 2-dice-throw example
Probability Mass Function
7/36
Probability
1/6
5/36
1/9
1/12
1/18
1/36
0
2
3
4
5
6
7
8
9
10 11 12
Range of Random Variable
E (X ) = 2
1
1
1
1
+3
+4
+ + 12
= 7 (make sense?).
36
18
12
36
Textbook sections: 3-4, 3-5, 2-1.4
Review Variance: 2-dice-throw example
Probability Mass Function
7/36
Probability
1/6
5/36
1/9
1/12
1/18
1/36
0
2
3
4
5
6
7
8
9
10 11 12
Range of Random Variable
E (X ) = 2
1
1
1
1
+3
+4
+ + 12
= 7 (make sense?).
36
18
12
36
V (X ) = E (X 2 ) [E (X )]2 =
1
1
1
22
+ 32
+ + 122
36
18
36
72 5.83.
Textbook sections: 3-4, 3-5, 2-1.4
Review Variance: 2-dice-throw example
Probability Mass Function
7/36
Probability
1/6
5/36
1/9
1/12
1/18
1/36
0
2
3
4
5
6
7
8
9
10 11 12
Range of Random Variable
E (X ) = 2
1
1
1
1
+3
+4
+ + 12
= 7 (make sense?).
36
18
12
36
V (X ) = E (X 2 ) [E (X )]2 =
1
1
1
22
+ 32
+ + 122
36
18
36
72 5.83. (X ) 2.42.
Textbook sections: 3-4, 3-5, 2-1.4
Examples (cont.)
Given a random variable X . Suppose its probability mass function is given as
follows.
81
f (x ) = ( )( )x , x = 1, 2, 3.
72
Calculate V (X ).
Textbook sections: 3-4, 3-5, 2-1.4
Examples (cont.)
Given a random variable X . Suppose its probability mass function is given as
follows.
81
f (x ) = ( )( )x , x = 1, 2, 3.
72
Calculate V (X ).
Solution: (Just use the formula of variance!)
E (X ) =
X
x
81
81
81
xf (x ) = 1 ( )( ) + 2 ( )( )2 + 3 ( )( )3 1.57.
72
72
72
V (X ) = E (X 2 ) [E (X )]2 =
Textbook sections: 3-4, 3-5, 2-1.4
Examples (cont.)
Given a random variable X . Suppose its probability mass function is given as
follows.
81
f (x ) = ( )( )x , x = 1, 2, 3.
72
Calculate V (X ).
Solution: (Just use the formula of variance!)
E (X ) =
X
x
81
81
81
xf (x ) = 1 ( )( ) + 2 ( )( )2 + 3 ( )( )3 1.57.
72
72
72
V (X ) = E (X 2 ) [E (X )]2 =
12 f (1) + 22 f (2) + 32 f (3) 1.572 1.43.
Textbook sections: 3-4, 3-5, 2-1.4
Discrete Uniform Distribution
Suppose that the range of X is {x1 , x2 , . . . , xn }. X has a discrete
uniform distribution if
1
f (xi ) = .
n
Textbook sections: 3-4, 3-5, 2-1.4
Discrete Uniform Distribution
Suppose that the range of X is {x1 , x2 , . . . , xn }. X has a discrete
uniform distribution if
1
f (xi ) = .
n
Example: the face value of a fair die throw.
Textbook sections: 3-4, 3-5, 2-1.4
Discrete Uniform Distribution
Suppose that the range of X is {x1 , x2 , . . . , xn }. X has a discrete
uniform distribution if
1
f (xi ) = .
n
Example: the face value of a fair die throw.
Q: Is the sum of the face values of two fair dice throw a discrete
uniform random variable?
Textbook sections: 3-4, 3-5, 2-1.4
Discrete Uniform Distribution (cont.)
Mean and Variance of Discrete Uniform Distribution
Suppose X is a discrete uniform random variable on consecutive
integers {a, a + 1, a + 2, . . . , b }. Then
E (X ) =
b+a
(b a + 1)2 1
, V (X ) =
.
2
12
Textbook sections: 3-4, 3-5, 2-1.4
Discrete Uniform Distribution (cont.)
Mean and Variance of Discrete Uniform Distribution
Suppose X is a discrete uniform random variable on consecutive
integers {a, a + 1, a + 2, . . . , b }. Then
E (X ) =
b+a
(b a + 1)2 1
, V (X ) =
.
2
12
Examples
E.g. 1. Let X denote the face value of a fair die throw. Calculate
E (X ) and V (X ).
Textbook sections: 3-4, 3-5, 2-1.4
Discrete Uniform Distribution (cont.)
Mean and Variance of Discrete Uniform Distribution
Suppose X is a discrete uniform random variable on consecutive
integers {a, a + 1, a + 2, . . . , b }. Then
E (X ) =
b+a
(b a + 1)2 1
, V (X ) =
.
2
12
Examples
E.g. 1. Let X denote the face value of a fair die throw. Calculate
E (X ) and V (X ).
E.g. 2 Let the random variable X have a discrete uniform
distribution on the integers 1 x 3. E (X ) =?, V (X ) =?
Textbook sections: 3-4, 3-5, 2-1.4
Binomial Distribution
To introduce binomial distribution, we need to learn counting
techniques rst.
Textbook sections: 3-4, 3-5, 2-1.4
Sec 2-1.4 Counting Techniques: Permutations
Example: White Collar
Suppose that Neal and Peter are trying to open a safe. They know
that the code to open the safe contains distinct 4 digits from 0 to
9. If Neal is to exhaust all the possible combinations, how many
times does Neal have to try?
Textbook sections: 3-4, 3-5, 2-1.4
Sec 2-1.4 Counting Techniques: Permutations
Example: White Collar
Suppose that Neal and Peter are trying to open a safe. They know
that the code to open the safe contains 4 distinct digits from 0 to
9. If Neal is to exhaust all the possible combinations, how many
times does Neal have to try?
Q: are 1234 and 4321 the same codes? A more general question:
does order matters?
Textbook sections: 3-4, 3-5, 2-1.4
Sec 2-1.4 Counting Techniques: Permutations
Example: White Collar
Suppose that Neal and Peter are trying to open a safe. They know
that the code to open the safe contains 4 distinct digits from 0 to
9. If Neal is to exhaust all the possible combinations, how many
times does Neal have to try?
Q: are 1234 and 4321 the same codes? A more general question:
10
does order matters? (A: 10 * 9 * 8 * 7, denoted as P4 )
Textbook sections: 3-4, 3-5, 2-1.4
Sec 2-1.4 Counting Techniques: Combinations
E.g. 1. How many possible ways are there to choose 3 items out 4
distinct items?
Textbook sections: 3-4, 3-5, 2-1.4
Sec 2-1.4 Counting Techniques: Combinations
E.g. 1. How many possible ways are there to choose 3 items out 4
distinct items?
Textbook sections: 3-4, 3-5, 2-1.4
Sec 2-1.4 Counting Techniques: Combinations
E.g. 1. How many possible ways are there to choose 3 items out 4
distinct items?
E.g.2. Suppose that a department has 10 faculty members. You
need to choose 4 faculty members to form a committee to guide
your study. How many dierent committees can you form?
Textbook sections: 3-4, 3-5, 2-1.4
Sec 2-1.4 Counting Techniques: Combinations
E.g. 1. How many possible ways are there to choose 3 items out 4
distinct items?
E.g.2. Suppose that a department has 10 faculty members. You
need to choose 4 faculty members to form a committee to guide
your study. How many dierent committees can you form?
Q: Is a committee consisting of Steve, Joe, Susan and Kay
dierent than that consisting of Kay, Susan, Joe and Steve?
More general question: does order matters?
Textbook sections: 3-4, 3-5, 2-1.4
Sec 2-1.4 Counting Techniques: Combinations
E.g. 1. How many possible ways are there to choose 3 items out 4
distinct items?
E.g.2. Suppose that a department has 10 faculty members. You
need to choose 4 faculty members to form a committee to guide
your study. How many dierent committees can you form?
Q: Is a committee consisting of Steve, Joe, Susan and Kay
dierent than that consisting of Kay, Susan, Joe and Steve?
10
More general question: does order matters? (A: P4 /4! = 210)
Textbook sections: 3-4, 3-5, 2-1.4
Permutations and Combinations
Permutation
Select r distinct elements from n distinct elements, and order
matters!
Prn = n (n 1) (n 2) (n r + 1) =
n!
.
(n r )!
Textbook sections: 3-4, 3-5, 2-1.4
Permutations and Combinations
Permutation
Select r distinct elements from n distinct elements, and order
matters!
Prn = n (n 1) (n 2) (n r + 1) =
n!
.
(n r )!
Combinations
Select r distinct elements from n distinct elements, and order
DOES NOT matter!
Crn
=
n
r
Pn
= r=
r!
n!
(nr )!
r!
=
n!
.
(n r )!r !
Textbook sections: 3-4, 3-5, 2-1.4
Permutations and Combinations: More Examples
E.g. 1 An encyclopedia has eight volumes. In how many ways
can the eight volumes be replaced on the shelf?
Textbook sections: 3-4, 3-5, 2-1.4
Permutations and Combinations: More Examples
E.g. 1 An encyclopedia has eight volumes. In how many ways
can the eight volumes be replaced on the shelf?
E.g. 2 Mr. Smith is the Chairman of a committee. In how
many ways can a committee of 5 be chosen from 10 people
(including Mr. Smith) given that Mr. Smith must be one of
them?
Textbook sections: 3-4, 3-5, 2-1.4
Permutations and Combinations: More Examples
E.g. 1 An encyclopedia has eight volumes. In how many ways
can the eight volumes be replaced on the shelf?
E.g. 2 Mr. Smith is the Chairman of a committee. In how
many ways can a committee of 5 be chosen from 10 people
(including Mr. Smith) given that Mr. Smith must be one of
them?
E.g. 3 Picking a President, Vice President, an accountant,
and an consultant from a group of 10 people.
Textbook sections: 3-4, 3-5, 2-1.4