230 test review

(Function, not precise).

A function assigns a single value to each argument. The set of possible arguments is called the domain, and the set of values is called the range. Example. f(x) = x^2 has domain R and range [0, ∞).

Undefined function

A function is said to be undefined, or "not defined" at points outside its domain. Example. The natural logarithm of x, ln(x) or loge (x), is not defined for negative numbers

Discrete Random Variable, Defn 2

A random variable X is discrete if its cdf FX(x) is a step function of x

Random Variable

A random variable is a function that assigns a real number to each outcome in the sample space of an experiment. Notation. We will denote random variables with upper case letters (usually at the end of the alphabet) such as X, Y , or Z. We reserve lower case letters such as x, y, and z to represent constants. Since random variables are functions, for a sample space S, we write X : S → R.

Replication

A replication is one instance of the random experiment, which results in exactly one outcome.

Discrete Sample Space

A sample space S is discrete if it is finite or countably infinite

Set

A set is a collection of items. Notation. Sets are usually denoted by capital letters, such as A, B, C, E, F, G, S. A = {x, y, z} denotes that A contains elements (or members) x, y, and z. If a set has members defined by a condition, A = {x : x satisfies the condition}, where the colon is read as "such that." MR (the book) uses the notation A = {x | x satisfies the condition}, where " | " is read as "such that." x ∈ A denotes that x "is an element of" A.

Axiom

An axiom is a statement that is assumed and requires no proof.

Remark 2

An event is a set. All of the set operators, such as union, intersection, and complement, operate on events because events are sets. Further, the distributive laws and DeMorgan's laws apply to events.

(Event).

An event is a subset of the sample space S. (That is, E ⊆ S). For a given replication of the experiment, the event E occurs if it contains the outcome; otherwise it does not occur. Remark 2 An event is a set. All of the set operators, such as union, intersection, and complement, operate on events because events are sets. Further, the distributive laws and DeMorgan's laws apply to events.

The Three 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 and satisfies the following three axioms: 1. P{S} = 1. (With probability 1, the outcome will be a point in the sample space S.) 2. 0 ≤ P{E} ≤ 1. (The probability that the outcome of the experiment is an outcome in E is a number between 0 and 1.) IE 230 Concise Notes · S. Hunter · August 28, 2014 · 3 of 223. For all mutually exclusive events E1 and E2, (that is, E1 ∩ E2 = ∅), we have P{E1 ∪ E2} = P{E1} + P{E2}. (If E1 and 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 and E2)

Result 3.8

(Properties of a cdf ‡ ). The function FX(x) is a cdf if and only if the following conditions hold: 1. limx→−∞ F(x) = 0 and limx→∞ F(x) = 1. 2. FX(x) is a nondecreasing function of x. 3. FX(x) is right-continuous, that is, for every number x0, limx↓x0 FX(x) = FX(x0)

Random Experiment

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

(Probability Mass Function (pmf

. For a discrete random variable, the probability mass function (pmf) is fX(x) = P{X = x} for every real number − ∞ < x < ∞. That is, the domain of fX(x) is R, and hence fX(x) is defined for all x ∈ R.

The following four statements are equivalent: 2

Events A and B are independent. • Events Ac and B are independent. • Events A and Bc are independent. • Events Ac and Bc are independent.

Baby Bayes' Theorem

For P{B} > 0, P{A | B} =(P{B | A}P{A})/P{B} .

Conditional Probability

For P{B} > 0, the conditional probability of A given B is P{A | B} = P{A ∩ B}/P{B}

DeMorgan's Laws

For any sets A and B, (A ∪ B)^c = A^c∩ B^c and (A ∩ B)^c = A^c ∪ B^c .

Distributive Laws

For any sets A, B, and C, it holds that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

Axioms Result 5: Always true

For any two events E1 and E2, P{E1 ∪ E2} = P{E1} + P{E2} − P{E1 ∩ E2}. Remark 4. Notice that this result can be generalized. For any three events, E1, E2, and E3, P{E1 ∪ E2 ∪ E3} =P{E1} + P{E2} + P{E3} − P{E1 ∩ E2} − P{E2 ∩ E3} − P{E1 ∩ E3} + P{E1 ∩ E2 ∩ E3} For n events, continue the pattern and alternate signs.

Axioms Result 1: Complement

For every event E, P{Ec} = 1 − P{E}. (In particular, P{∅} = 1 − P{S} = 0, which implies that the "impossible" event has probability zero.

Axioms Result 2: Dominance

If E1 ⊂ E2, then P{E1} ≤ P{E2}.

Result 3.6.

If a ≤ b, then FX(a) ≤ FX(b).

Subset

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

Superset

If all members of a set A are contained in a set B, then B is a superset of A, written B ⊇ A.

(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 events E1, E2, . . . , En are mutually exclusive, then P{∪n i=1Ei} = Pn i=1 P{Ei}

Partition of the Sample Space

If ∪ n i=1Ei = S and E1, E2, . . . , En are mutually exclusive, then E1, E2, . . . , En are said to partition the sample space.

The Law of Total Probability

Let E1, E2, . . . , En be mutually exclusive events such that ∪ n i=1 Ei = S. Then P{A} = Pn i=1 P{A ∩ Ei} = Pn i=1 P{A | Ei}P{Ei}. look at notes

(Discrete Random Variable, Defn 1

Let S be a sample space. A discrete random variable is a function X : S → R that takes on a finite number of values or a countably infinite number of values. Remark 8. Discrete random variables often arise from counting

(Multiplication Rule

P{A ∩ B} = P{A | B}P{B}.

Mutually Independent Events

See notes The n events A1, A2, . . . , An are mutually independent if and only if for every subset Ai1 , Ai2, . . . , Aik of the n events, P{Ai1 ∩ Ai2 ∩ . . . ∩ Aik} = P{Ai1}P{Ai2} · · · P{Aik} for k = 2, 3, . . . , n. That is, P∩k j=1 Aij = Qk j=1 P{Aij }. Remark 6 (Pairwise Independence). Pairwise independence is a weaker form of independence than mutual independence, which requires only that every pair of events be independent (k = 2 in the Definition 2.24).

(Cardinality

The cardinality of a set A, written |A|, is the number of elements in the set.

Complement

The complement of a set A is the set of items not in A. Notation. Ac = {x : x /∈ A} or A0 = {x : x /∈ A}

Cumulative Distribution Function (cdf)‡

The cumulative distribution function (cdf) of any random variable X is FX(x) = P{X ≤ x} for every real number − ∞ < x < ∞.

empty set

The empty set, denoted ∅, contains no items. It is the smallest set.

Intersection

The intersection of sets A and B is the set of items contained in both sets. Notation. A ∩ B = {x : x ∈ A and x ∈ B}. Remember A ∩ B means "A and B." Intersection can be extended to a collection of sets. Consider sets A1, A2, . . . , An, all subsets of the universe S. Then ∩ n i=1 Ai = {x ∈ S : x ∈ Ai for all i}

Probability Distribution

The probability distribution of a random variable x is a description, in whatever form, of the likelihoods associated with the values of x. Remark 7. Events can be constructed from random variables. Consider the case of rolling two dice, and define X as the sum of the numbers showing on the two dice. "X = 2" denotes the event (1, 1) is rolled. "X < 3" denotes the event (1, 1) is rolled. "X ≤ 3" denotes the event (1, 1) or (1, 2) or (2, 1) is rolled. Therefore stating P{X = 2} makes sense because P{·} is a set function and"X = 2" is an event, and events are sets. The statement P{X} is meaningless because X is a random variable, not a set.

Probability

The probability of an 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]. Remark 3. A common interpretation of probability is as a relative frequency: if the experiment were repeated infinitely often, P{E} is the fraction of the replications in which E occurs.

(Countably Infinite

The set A is countably infinite |A| is infinite, but its members can be counted, that is, a unique integer can be assigned to each member.

Finite

The set A is finite if |A| is finite.

Uncountably Infinite

The set A is uncountably infinite if |A| is infinite and its members cannot be counted. (E.g., R is uncountably infinite

Sample Space

The set S of all possible outcomes of a particular random experiment is called the sample space. (Choose the simplest such space to answer the question at hand.)

Universe

The set containing all relevant items is called the universe. It is the largest set.

(Partition)

The sets B1, B2, . . . , Bn partition the set A if ∪ n i=1 Bi = A and Bi ∩Bj = ∅ for all i 6= j. That is, together, all the Bi 's contain all the elements of A, and each member of A lies in exactly one of the Bi 's. (The Bi 's are "mutually exclusive and collectively exhaustive" with respect to the set A.)

Support‡

The support of a distribution is the set of all x ∈ R such that fX(x) > 0. That is, X = {x ∈ R : fX(x) > 0}. The support X is written with the distribution function, and the function is assumed to be zero elsewhere. In this class, we usually state explicitly that the function is zero elsewhere.

Union

The union of sets A and B is the set of items contained in at least one of the sets. Notation. A ∪ B = {x : x ∈ A or x ∈ B}. Remember A ∪ B means "A or B or both." Union can be extended to a collection of sets. Consider sets A1, A2, . . . , An, all subsets of the universe S. Then ∪ n i=1 Ai = {x ∈ S : x ∈ Ai for some i}.

Independence

Two events A and B are independent if P{A ∩ B} = P{A}P{B} Remark 5. For independent events A and B, P{A | B} =P{A ∩ B}/P{B} =P{A}P{B}/P{B} = P{A}.

Mutually Exclusive / Disjoint

Two events, say E1 and E2, are mutually exclusive (or disjoint) if they cannot occur together in the same replication of the experiment, that is, E1 ∩ E2 = ∅. More generally, E1, E2, . . . , En are mutually exclusive if only one can occur in the same replication, that is, Ei ∩ Ej = ∅ for every pair of events.

Set Equality

Two sets A and B are equal, and we write A = B, if they contain the same elements. (That is, A ⊆ B and A ⊇ B.) Notation. The symbols ⊂ and ⊃ are "strict" versions of ⊆ and ⊇, just like < and > are "strict" versions of ≤ and ≥. That is, A ⊂ B if all elements of A are contained in B and A and B are not equal. Notice that not all texts make this distinction.

(Mutually Exclusive / Disjoint

Two sets A and B are mutually exclusive (or disjoint) if they contain no elements in common, that is, A ∩ B = ∅. More generally, A1, A2, . . . , An are mutually exclusive if Ai ∩ Aj = ∅ for every pair of sets where i 6= j.

Set Operators

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

(Unconditional Probability

With respect to a sample space S, P{A | S} = P{A ∩ S}/P{S}:= P{A} is the unconditional or marginal probability of A.

Result 3.5

see notes For a discrete random variable X having possible values x1, x2, . . . , xm, the cdf is P{X ≤ x} = FX(x) = P all xi≤x fX(xi) for every real number − ∞ < x < ∞.

Result 3.7

‡For every random variable X, if a ≤ b, then P{a < X ≤ b} = FX(b) − FX(a).

The following four statements are equivalent:

• Events A and B are independent. • P{A ∩ B} = P{A}P{B} • P{A | B} = P{A} • P{B | A} = P{B}