Unit 4 Review

Probability

Probability

a chance process that is a number between 0 and 1

Simulaiton

imitation of a chance behavior, based on a model that accurately reflects

Performing of a Simulation

1. State: Ask a question of interest

2. Plan: Describe how to use chance to imitate one reptation of the process

3. Do: Perform many repetitions of the simulation

4. Conclude: Use results of simulation to answer question of interest

Sample space S

the set of all possible outcomes

Probability model

a description of some chance process that consists of a sample space S and a probability for each outcome

Event

is any collection of outcomes from some chance process; a subset of the sample space

Basic rules of probability

Probability is always a number from 0 to 1, if S is the sample space in a probability model P(S)=1), and P(A) = the number of outcomes corresponding to event A / the total number of outcomes in the sample space

Complement rule

P(A^c) = 1 - P(A)

Addition rule for mutually exclusive events

P(A or B) = P(A) + P(B)

Mutually exclusive (disjoint)

if they have no outcomes in common and so they can never occur together, P(A and B) = 0

General Addition Rule

P(A or B) = P(A) + (B) - P(A and B)

Conditional Probability

the probability that one event happens given that another event is already known to have happened; P(A|B) = P (A and B) / P(B)

Independent

if the occurrence of one event does not change the probability that the other event will happen

General Multiplication Rule

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

Multiplication Rule for Independent Events

P (A and B) = P(A) x P(B)

Random variable

a variable whose value is a numerical outcome of a random phenomenon

Discrete Random Variable

has a countable number of possible values

Continuous Random Variable

takes all values in an intervale of numbers and is measurable

Mean of X

𝝁𝒙 = 𝑬(𝒙) = ∑𝒙𝒊∙ 𝒑𝒊 = 𝒙𝟏 ∙ 𝒑𝒊 + 𝒙𝟐 ∙ 𝒑𝟐 + ⋯ + 𝒙𝒌 ∙ 𝒑𝒌

Conditions for Binomial Probability Distribution

Binary - each observation is a success or failure

Independence - observations must be intendent, the result of one does not affect another

Number - procedures has a fixed number of trials (n)

Success - probability of success (p)

Binomial Probability Formula

P (X = k) = n! / k! (n-k)! (p)^k (1-p) ^n-k

Binompdf

the probability distribution function and determines P(X=k)

Binomcdf

a cumulative distribution function and determines P (X < or = k)

Mean (expected value) of a Binomial Random Variable

=np

Geometric probability distribution

Binary - each observation is either a success or failure

Independence - the observations must be independent, result of one does not affect another

Trials - variable of interest is the number of trials required to obtain the first success

Success - probability of success (p)

Geometric Probability Formula

P(X=n) - (1-p)^n-1(p)

Geometpdf

the probability distribution function and determines P(X=n)

Geometcdf

a cumulative distribution function and determines P(X , or = n)

Mean (expected value) of a Geometric Random Variable

= 1/p