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|)