Class 2

Probability Theory

Discrete Random Variables

Expectations

Introduction to Probability• A probability is a number between 0 and 1 inclusive

that measures the likelihood of the occurrence of an event.

• Given an event, E, we will write the probability of E as P{E} or Pr{E}.

• One early notion of probability came from the observation of relative frequencies of events that occurred from replication of some process (like rolling dice).

Outcome Spaces• Another perspective on probabilities comes from the

notion of an outcome space. • Consider all possible outcomes (events) of an experiment.• Example: Consider the experiment performed by throwing a die

and looking at the top surface. What is the outcome space?

• What is the P{3}? P{odd number}?

Applying Set Concepts to Compute Probabilities

• The union of events A and B (AB) are all of the outcomes that make A or B occur.

• The intersection of events A and B (AB) are all of the outcomes that make A and B occur at the same time.

• For the die example, let A: {throw less than or equal to 4} and B: {even number}.

Example Computation (cont.)

• What is P{A}?

• P{B}?

• P{AB}?

• P{AB}?

It turns out that

P{AB} = P{A} + P{B} - P{AB} .

Conditional Probability

• Consider all families with two children. What does the outcome space look like?

• Select such a family. Let A:{the family has at least one male child} and B:{the family has exactly two male children}.

• What is P{A}? P{B}? P{AB}?

Conditional Probability (cont.)

• Now suppose that A has occurred. What had to happen?

• Are these outcomes equally likely?

• We will write {B|A} to indicate the event B given that A has occurred. What is P{B|A}?

It turns out that P{B|A} = P{AB}/P{A}

Special Relationships

• Two events, A and B, are said to be mutually exclusive if and only if P{AB}=0. In this case, P{AB} = P{A} + P{B}.

• Two events, A and B, are said to be independent if and only if P{A|B} = P{A}. In this case, P{AB} = P{A}P{B}.

• An event B is said to be the complement of A if it always happens exactly when A does not happen. In this case P{B} = 1 - P{A}.

Contingency Tables

• A contingency table is a cross classification of data displayed in a tabular form.

• In a decision making context (such as a marketing study), the decision maker might treat the relative frequencies found in the table as if they were probabilities.

• Consider the attached example where a department store has analyzed 10,000 sales to better understand the relationship between type of purchase (cash or credit) and merchandise.

Contingency Tables

AB:

AB:

BC:

BC:

A|B:

Contingency Tables

P{AB}

P{AB}

P{BC}

P{BC}

Contingency Tables

P{A|B}

Random Variables

• A random variable is a model of a population. When we discuss random variables, we are simply talking about populations.

• To illustrate how a population can be modeled, consider the experiment where we flip a coin three times.

Populations and RV’s

1

1

11

1

1

1

1

3

1

1

2

2 2

2

2

2

2

2

2

20

33

3

3

0

00

0

0

How large is the population?

Count the number of heads that you see.

Probability Distributions

• The probability distribution of X is just a listing (or graph) of the values of X along with the probability that X assumes that value.

X p(x)0 1/81 3/82 3/83 1/8

Probability Distribution - Graph

Probability Distribution

1/8

3/8 3/8

1/8

0.00

0.10

0.20

0.30

0.40

0 1 2 3

Values of X

Pro

babi

lity

• Compute:– P{X1}

– P{X>2}

– P{X is odd}

Random Variables

• Remember:– A probability distribution is a list of the values and

probabilities that a random variable assumes.

– These values can be thought of as the values in a population, and the probabilities as the proportion of the population that a specific value makes up.

• Random variables can be classified as being discrete or continuous. Continuous random variables assume values along a continuum.

Discrete Random Variables

• Our example of counting the number of heads in three coin flips was an example of a discrete random variable. Why?

• An example of a continuous random variable is W = the amount of gasoline in your car.

• We will return to continuous random variables shortly.

The Mean of a Random Variable

• Since a random variable is just describing a population, it has a mean (average) value. What should we call this average?

xxxp

XE

)(

)(

Variance of a Random Variable

• The variance provided a measure of dispersion of the values in the data, the average squared distance from .

x

xpx )()( 22