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MTH 365 midterm 1 study
key definitions and theorems from chapters 1 and 2 of Hogg and Tanis
23
Mathematics
Undergraduate 3
02/06/2014

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Term
probability mass function (p.m.f.)
Definition

The probability mass function (p.m.f.) f(x) of a discrete random variable X is a function that satisfies the following properties:

(a) f(x) > 0, if x ε S

(b) Σf(x) = 1, for x ε S

(c) P(X ε A) = Σf(x), for x ε A where A is a proper subset of S 

 

Term
random variable
Definition

Given a random experiment with an outcome space S, a function X that assigns one and only one real number X(s)=x to each element in S is called a random variable.

The space of X is the set of real numbers {x: X(s)=x, s ε S}, where s ε S means that s belongs to the set S.

Term

mathematical expectation

(expected value)

Definition

If f(x) is the p.m.f. of the random variable X of the discrete type with space S, and if the summation Σu(x)f(x) for x ε S exists, then the sum is called the mathematical expectation or expected value of the function u(X), and it is denoted by E[u(X)]. That is

 E[u(X)] = Σu(x)f(xfor x ε S

Term

properties of mathematical expectation

Definition

When it exists, the mathematical expectation E satisfies the following properties: 

(a) If c is a constant, then E(c)=c.

(b) If c is a constant and u is a function, then

E[cu(X)]=cE[u(X)].

(c) If c1 and c2 are constants and u1 and u2 are functions, then E[c1u1(X) + c2u2(X)] = c1E[u1(X)] + c2E[u2(X)] 

Term
De Morgan's Laws
Definition

(A ∪ B)' = A' ∩ B'

 

(A ∩ B)' = A' ∪ B'

Term
commutative laws for set algebra
Definition

A ∪ B = B ∪ A

 

A ∩ B = B ∩ A 

Term
associative laws for set algebra
Definition

 (A ∪ B) ∪ C = A ∪ (B ∪ C)

 

(A ∩ B) ∩ C = A ∩ (B ∩ C) 

Term
distributive laws for set algebra
Definition

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

 

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 

Term
properties of a binomial experiment
Definition

1. A Bernoulli (success-failure) experiment is performed n times.

2. The trials are independent.

3. The probability of success on each trial is a constant p; the probability of failure is q = 1 - p.

4. The random variable X equals the number of successes in the n trials.

Term
mean, variance and standard deviation of a random variable with a binomial distribution
Definition

μ = np

σ= np(1-p) or σ= npq

σ = √(np(1-p)) or σ √(npq)

Term
mean of a random variable X
Definition
μ = E(X) = Σxf(x), x ε S
Term
variance of a random variable X
Definition
σ2 = Σ(x - μ)2f(x), x ε S = E(X2) - [E(X)]2
Term
conditional probability
Definition

The conditional probability of an event A, given that event B has occurred, is defined by

P(A|B) = P(AB)/P(B)

provided that P(B) > 0.

Term
multiplication rule
Definition

The probability that two events, A and B, both occur is given by the multiplication rule:

P(A ∩ B) = P(A)P(B|A)

or by

P(B ∩ A) = P(B)P(A|B).

Term

addition rule for two events

(Hogg and Tanis theorem 1.2-5)

Definition

The probability that either (or both) of two events, A and B, occur is given by the addition rule:

P(A U B) = P(A) + P(B) - P(A ∩ B).

proof hint 1: Rewrite A U B as the union of mutually exclusive events: A U (A' ∩ B).

proof hint 2: Do that trick again: B = (A ∩ B) U (A' ∩ B)

Term
probability (definition)
Definition

Probability is a real-valued set function P that assigns, to each event A in the sample space S, a number P(A), called the probability of the event A, such that the following properties are satisfied:

(a) P(A) ≥ 0,

(b) P(S) = 1, 

(c) If A1, A2, A3, ... are events and Ai ∩ A= Ø, i ≠ j, then

P(A1A2 U ... U Ak) = P(A1) + P(A2) + ... + P(Ak)

for each positive integer k, and

P(A1 U A2 U AU ...) = P(A1) + P(A2) + P(A3) + ...

for an infinite, but countable, number of events.

Term

probability of the complement event A

(Hogg and Tanis theorem 1.2-1)

Definition

For each event A, P(A) = 1 - P(A')

Proof hint: S = A + A'

Term

probability of Ø

(Hogg and Tanis theorem 1.2-2)

Definition

P(Ø) = 0

Proof hint: Let A = Ø then A' = S, then apply probability of inverse theorem

Term

relationship between P(A) and P(B) if A is a subset of B

(Hogg and Tanis theorem 1.2-3)

Definition

If events A and B are such that A is a subset of B, then P(A) ≤ P(B

Proof hints: start with B = A U (B ∩ A'). Remember that for any event A, P(A) ≥ 0.

Term

upper limit of probability of event A

(Hogg and Tanis theorem 1.2-4)

Definition

P(A) ≤ 1

Proof hint: A is a subset of S

Term

addition rule for three events

(Hogg and Tanis theorem 1.2-6)

Definition

The probability that at leat one of three events, A, B or C, occur is given by the addition rule:

P(A U C) = P(A) + P(B) + P(C) - P(A ∩ B)- P(A ∩ C) -  P(B ∩ C) + P(A ∩ B ∩ C)

proof hint: A U C = A U (C)

Term
independent events
Definition

Events A and B are independent if and only if P(A ∩ B) = P(A)P(B). Otherwise A and B are called dependent events.

(Note that this is equivalent to saying P(B|A) = P(B).)

Term

independence of events and complements

(Hoggand Tanis theorem 1.5-1)

Definition

If A and B are independent events, then the following pairs of events are also independent:

(a) A and B',

(b) A' and B,

(c) A' and B'

proof hint for (a): P(B' |A) =  1 - P(B|A

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