IE 230 exam 1

Following statements are equivalent

1.) events A & B are independent 2.) P(A ∩ B) = P(A)*P(B) 3.) P(A | B) = P(A) = P(A | S) 4.) P(B | A) = P(B) • if info about A tells me nothing about B, vice versa

More equivalent statements

1.) events A & B are independent 2.) events A^c & B are independent 3.) events A & B^c are independent 4.) events A^c & B^c are independent

A = {x: x satisfies the condition}

A set that has members (or elements) defined by a condition

Axiom Result 5 Generalized

For any 3 events E1, E2, & E3, P(E1 E2 E3) = P(E1) + P(E2) + P(E3) subtract middle 3x - P(E1 ∩ E2) - P add back once + P(E1 ∩ E2 ∩ E3) for n events, continue pattern & alternate signs

Axioms Result 5: Always True

For any two events E1 & E2, P(E1 ∪ E2) = P(E1) + P(E2) - P(E1 ∩ E2)

Bayes' Theorem

For events A & B where P(B) > 0, P(A | B) = {P(B | A) * P(A)}/P(B)

Undefined Function

Function is considered undefined at points outside of its domain

Discrete Random Variable

Function that takes on a countable number of values

3 Axioms of Probability Consider an experiment with sample space S. For each event E of the sample space S, we assume that a number P(E) is defined that satisfies...

I.) P(S) = 1 • with probability 1, the outcome will be a point in the sample space S II.) 0 ≤ P(E) ≤ 1 • the probability that the outcome of the experiment is an outcome of E is between 0 & 1 III.) P(E1 ∪ E2) = P(E1) + P(E2) for all mutually exclusive events E1 & E2 • if E1 & E2 have no events in common, then the relative frequency of outcomes in E1 ∪ E2 is the sum of the relative frequencies of the outcomes in E1 & E2

Axioms Result 2: Dominance

If E1 ⊆ E2, then P(E1) ≤ P(E2)

Axioms Result 4: Equally Likely Events

If equally likely events E1, E2, ..., En partition the sample space, then P(Ei) = 1/n for i = 1, 2 , ..., n

Axioms Result 3: Axiom 3 for n Events

If the events are mutually exclusive, you can add all of their probabilities

Conditional Probability

Let A & B be events in the sample space S. For P(B) > 0, conditional probability of A given B is P(A | B) = P(A ∩ B)/P(B)

Law of Total Probability

Let E1, E2, ..., En be mutually exclusive events such that these events partition the sample space S & are collectively exhaustive, then P(A) = sum of P(A ∩ Ei) = sum of P(A | Ei)*P(Ei)

Multiplication Rule

P(A ∩ B) = P(A | B)*P(B) = P(B | A)*P(A) switch roles of A & B in conditional probability

Probability Distribution

Probability distribution of a random variable X is a description, in whatever form, of the likelihoods associated with the values of X

Random Experiment

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

Continuous Sample Space

Sample space S is continuous if it contains an interval of real numbers (always uncountable)

Discrete Sample Space

Sample space S is discrete if is countable

Countable

Set is countable if its members can be counted, meaning a unique integer can be assigned to each member

Finite

Set is finite (or infinite) if its cardinality is finite (or infinity)

Set Operators

Union, intersection, & complement are operations that are defined for sets

Pairwise Independence

Weaker form of independence, requires only that every pair of events be independent (k = 2)

Unconditional Probability

With respect to sample space S, P(A | S) = P(A ∩ S)/P(S) = P(A) is the unconditional probability or marginal probability of event A

x ∈ A

x is an element of A

DeMorgan's Laws For any sets A & B: • (A U B)^c = • (A ∩ B)^c =

• (A U B)^c = A^c ∩ B^c • (A ∩ B)^c = A^c U B^c

Distributive Laws For any sets A, B, & C: • A ∩ (B U C) = • A U (B ∩ C) =

• A ∩ (B U C) = (A ∩ B) U (A ∩ C) • A U (B ∩ C) = (A U B) ∩ (A U C)

Function

• Assigns a single value to each argument • Domain: set of possible arguments • Range: set of values

Set

• Collection of items • Denoted by a capital letter

Complement

• Complement of set A is the set of items NOT in A • A^c = {x: x ∉ A}

Empty set

• Contains no items • Smallest set • Denoted Ø • A subset of every set, including the empty set itself

Axioms Result 1: Complement

• For every event E, P(E^c) = 1 - P(E) • P(Ø) = 1 - P(S) = 0, ∴ the "impossible" event has probability 0

Random Variable

• Function that assigns a real number to each outcome in the sample space of an experiment • Denoted as a capital letter near the end of the alphabet (X, Y, Z)

Partition

• If U^n, i = 1 Ei = S & E1, E2, ..., En are mutually exlusive, then E1, E2, ..., En partition the sample space • Have to split it up into mutually exclusive events (no overlap) & include the entire sample space

Subset

• If all members of a set A are contained in set B, then A is a subset of B • Denoted: A ⊆ B

Cardinality

• Number of elements in a set • Denoted |A|

Replication

• One instance of the random experiment • Results in exactly one outcome

Sample Space

• Set S of all possible outcomes of a particular random experiment • Choose simplest sample space • Sample space S is our universe

Universe

• Set containing all relevant items • Denoted S

Union

• Set of items contained in at least one of the sets • A or B or both • A ∪ B = {x : x ∈ A or x ∈ B}

Intersection

• Set of items contained in both sets • A & B • A ∩ B = {x: x ∈ A & x ∈ B}

Mutually Exclusive/Disjoint (sets)

• Sets that have no elements in common • A ∩ B = Ø

Event

• Subset of sample space S • For a given replication of the experiment, event E occurs if it contains the outcome, otherwise it does not occur • Events are sets

Independence

• Two events A & B are independent if P(A ∩ B) = P(A)*P(B) • NEVER ASSUME independence unless explicitly told so in problem

Mutually Exclusive/Disjoint (events)

• Two events E1 & E2 are mutually exclusive if they cannot occur together in the same replication of the experiment • E1 ∩ E2 = Ø • Example: E1 = {heads} E2 = {tails} (no overlap)

Equal

• Two sets A & B are equal if they contain exactly the same elements • Denoted A = B • (∴ A ⊆ B & B ⊆ A)

Mutually Independent Events

• n events A1, A2, ..., An are mutually independent if & only if every subset Ai1, Ai2, Aik of the n events P(Ai1 ∩ Ai2 ∩ ... ∩ Aik) = P(Ai1)*P(Ai2)*...*P(Aik) for k = 2, 3, ..., n • Very strong assumption that everything is independent

Probability

• of event E, denoted P(E), is a numerical measure of how likely the event E is to occur when the experiment is performed • P(•) is a function that maps a set to a real number in [0,1] • Relative frequency: if the experiment were repeated infinitely often, P(E) is the fraction of replications in which E occurs