IE 230 Exam 1

Results of Indepedence

1. Events A and B are independent 2. Events Ac and B are independent 3. Events A and Bc are independent

Independence

1. Events A and B are independent 2. P(A and B) = P(A)P(B) 3. P(A|B) = P(A) 4. P(B|A) = P(B)

Equal

A C B, B C A

Intersection

A and B

Random variable

A function that assigns a real number to each outcome in the sample space of an experiment. All random variables are capital letters at the end of the Alphabet. constants are lowercase variables.

Random experiment

A procedure that can result in a different outcome each time it is performed

Continuous sample space

A sample space is continuous if it contains an interval of real numbers

Discrete sample space

A sample space is discrete if its countable

Function

Assigns a single value to each argument

second axiom of probability

0 <= P(E) <= 1. The probability that the outcome of the experiment in a outcome in E is a number between 0 and 1.

Law of Total Probability

Events have to be mutually exclusive and collectively exhaustive. P(D) = P(D and W) + P(D and Wc). P(D) = P(D|W)P(W) + P(D|Wc)P(Wc)

third axiom of probability

For all mutually exclusive events, P(E1 U E2) = P(E1) + P(E2)

axiom result 5: Always Ture

For any two events, P(E1 U E2) = P(E1) + P(E2) - P (E1 and E2)

Partition

If E1, E2, En are mutually exclusive they are said to partition the sample space.

Event

If a subset of the sample space. Even occurs if it contains the outcome E C S

Subset

If all members of a set A are in contained in a set B, then A is a subset of B, A C B

axiom result 4: Equally likely events

If equally likely events partition the sample space than P(Ei) = 1/n

Pairwise vs mutual independence

Mutual independence if none of the events are shared, pairwise only requires that every pair be independent

mutually exclusive (disjoint)

No elements in common. A and B = empty set.

Replication

One instance of the random experiment, which results in exactly one outcme

Multiplication Rule

P(A and B) = P(A|B)*P(B) = P(B|A)P(A).

Conditional Probability

P(A|B) = P(A and B) / P(B)

Multiplication Rule 2

P(B|A) = P(B and A)/P(A) = P(A and B)/P(B)

first axiom of probability

P(S) = 1 (With probability one the outcome will be a point in the sample set S)

Domain

Set of possible arguments

Axiom

Statement that is assumed and requires no proof

Probability

The probability of an event E is a numerical measure of how likely the event E is to occur when the experiment is performed (has to be a real number between 0 and 1)

Sample Space

The set (S) of all possible outcomes of a particular random experiment

Mutually Exclusive/ disjoint events

Two events (E1 and E2) are mutually exclusive (disjoint) if they cannot occur together in the same replication of the experiment (E1 and E2 = empty set)

Set Operators

Union, intersection, and complement are operations defined for sets

Undefined function

a function is said to be undefined at points outside it domain.

axiom result 1: complement

for every event, P(Ec) = 1 - P(E). Impossible that the event has probability zero

axiom result 2: dominance

if E1 C E2, then P(E1) <= P(E2)

Union

in A, in B, or both

Range

set of values

Cardinality

the number of elements in a set (written |A|)