Expected Value and Variance - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CSci 2011

Discrete Mathematics

Lecture 24

ch 6.

Expected Value and Variance

Expected Value

The expected value of the random variable X(s) on

the sample space S is equal to

E(X) = ∑s ∈ S p(s) X(s)

Expected value of a Die

 X is the number that comes up when a die is rolled.  What is the expected value of X?  E(X) = 1/6 1 + 1/6 2 + 1/6 3 + … 1/6 6 = 21/6 = 7/

Three times coin flipping example

 X: number of heads  E(X) = 1/8 3 + 3/8 2 + 3/8 1 + 1/8 0 = 12/8 = 3/

Little More…

If X is a random variable and p(X=r) is the

probability that X=r, then

 E(X) = ∑r ∈ X(S) p(X=r) r.

What is the expected value of sum of the numbers

that appear when a pair of fair dice is rolled?

 p(X=2) = p(X=12) = 1/  p(X=3) = p(X=11) = 2/  p(X=4) = p(X=10) = 3/  p(X=5) = p(X=9) = 4/  p(X=6) = p(X=8) = 5/  p(X=7) = 6/  E(X) =

Ch. 8.

Relations

What is a relation

 Relation generalizes the notion of functions.

 Recall: A function takes EACH element from a set and maps it

to a UNIQUE element in another set  f: X → Y  ∀ x ∈ X, ∃ y such that f(x) = y

 Let A and B be sets.

A binary relation R from A to B is a subset of A × B  Recall: A x B = {(a, b) | a ∈ A, b ∈ B}  aRb: (a, b) ∈ R.

 Application

 Relational database model is based on the concept of relation.

More relation examples

Another relation example:

 Let A be the cities in the US  Let B be the states in the US  We define R to mean a is a city in state b  Thus, the following are in our relation: (Minneapolis, MN) (Philadelphia, PA) (Portland, MA) (Portland, OR) etc…

Most relations we will see deal with ordered pairs of

integers

Representing relations

CS

CS

CS

Alice

Bob

Claire

Dan

CS1901 CS2011 CS

Alice (^) X

Bob (^) X X

Claire

Dan (^) X X

We can represent

relations graphically:

We can represent relations in a table:

Not valid functions!

Relations on a set

A relation on the set A is a relation from A to

A

In other words, the domain and co-domain are

the same set

We will generally be studying relations of this

type

Relations on a set

Let A be the set { 1, 2, 3, 4 }

Which ordered pairs are in the relation

R = { ( a,b ) | a divides b }

R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)}

R 1 2 3 4

1 X X X X

2 X X

3 X

4 X

Relation properties

Six properties of relations we will study:

Reflexive

Irreflexive

Symmetric

Asymmetric

Antisymmetric

Transitive

Reflexivity vs. Irreflexivity

Reflexivity

 Definition: A relation is reflexive if ( a , a ) ∈ R for all a ∈ A

Irreflexivity

 Definition: A relation is irreflexive if ( a , a ) ∉ R for all a ∈ A

Examples

 Is the “divides” relation on Z +^ reflexive?  Is the “” (not ⊆) relation on a P(A) irreflexive?

irreflexive x o o x x

reflexive o x x o o

= < > ≤^ ≥

Symmetry, Asymmetry, Antisymmetry

A relation is symmetric if

 for all a, b ∈ A, ( a , b ) ∈ R ⇒ ( b , a ) ∈ R

A relation is asymmetric if

 for all a, b ∈ A, ( a , b ) ∈ R ⇒ ( b,a ) ∉ R

A relation is antisymmetric if

 for all a, b ∈ A, (( a , b ) ∈ R ∧ ( b,a ) ∈ R) ⇒ a = b  (Second definition) for all a, b ∈ A, (( a , b ) ∈ R ∧ a ≠ b ) ⇒ ( b,a ) ∉ R )

antisymmetric o o o o o x

asymmetric o o x x x x

symmetric x x o x x o

< > = ≤ ≥ isTwinOf

Notes on *symmetric relations

A relation can be neither symmetric or

asymmetric

R = { (a,b) | a=|b| }

This is not symmetric

-4 is not related to itself

This is not asymmetric

4 is related to itself

Note that it is antisymmetric