Lecture 6 Dustin Lueker. Standardized measure of variation ◦ Idea A standard deviation of 10 may indicate great variability or small variability,

STA 291Summer 2010

Lecture 6Dustin Lueker

Coefficient of Variation Standardized measure of variation

◦ Idea A standard deviation of 10 may indicate great

variability or small variability, depending on the magnitude of the observations in the data set

CV = Ratio of standard deviation divided by mean◦ Population and sample version

2STA 291 Summer 2010 Lecture 6

Example Which sample has higher relative

variability? (a higher coefficient of variation)◦ Sample A

mean = 62 standard deviation = 12 CV =

◦ Sample B mean = 31 standard deviation = 7 CV =

STA 291 Summer 2010 Lecture 6 3

Probability Terminology Experiment

◦ Any activity from which an outcome, measurement, or other such result is obtained

Random (or Chance) Experiment◦ An experiment with the property that the outcome cannot

be predicted with certainty Outcome

◦ Any possible result of an experiment Sample Space

◦ Collection of all possible outcomes of an experiment Event

◦ A specific collection of outcomes Simple Event

◦ An event consisting of exactly one outcome


Experiment, Sample Space, Event


Examples:Experiment1. Flip a coin2. Flip a coin 3 times3. Roll a die4. Draw a SRS of size

50 from a population

Sample Space

1.2.3.4.

Event1.2.3.4.

Complement Let A denote an event Complement of an event A

◦ Denoted by AC, all the outcomes in the sample space S that do not belong to the event A

◦ P(AC)=1-P(A)

Example◦ If someone completes 64% of his passes, then

what percentage is incomplete?


SA

Union and Intersection Let A and B denote two events Union of A and B

◦ A ∪ B◦ All the outcomes in S that belong to at least one

of A or B Intersection of A and B

◦ A ∩ B◦ All the outcomes in S that belong to both A and B


Additive Law of Probability Let A and B be two events in a sample

space S◦ P(A∪B)=P(A)+P(B)-P(A∩B)


S

A B

Additive Law of Probability Let A and B be two events in a sample

space S◦ P(A∪B)=P(A)+P(B)-P(A∩B)

At State U, all first-year students must take chemistry and math. Suppose 15% fail chemistry, 12% fail math, and 5% fail both. Suppose a first-year student is selected at random, what is the probability that the student failed at least one course?


Disjoint Events (Mutually Exclusive) Let A and B denote two events A and B are Disjoint (mutually exclusive)

events if there are no outcomes common to both A and B◦ A∩B=Ø

Ø = empty set or null set

Let A and B be two disjoint events in a sample space S◦ P(A∪B)=P(A)+P(B)


S

A B

Conditional Probability

◦ Note: P(A|B) is read as “the probability that A occurs given that B has occurred”

11

( )( | ) , provided ( ) 0

( )

P A BP A B P B

P B

STA 291 Summer 2010 Lecture 6

Assigning Probabilities to Events The probability of an event occurring is

nothing more than a value between 0 and 1◦ 0 implies the event will never occur◦ 1 implies the event will always occur

How do we go about figuring out probabilities?


Assigning Probabilities to Events Can be difficult Different approaches to assigning probabilities to

events◦ Subjective◦ Objective

Equally likely outcomes (classical approach) Relative frequency


Relies on a person to make a judgment on how likely an event is to occur◦ Events of interest are usually events that cannot

be replicated easily or cannot be modeled with the equally likely outcomes approach As such, these values will most likely vary from

person to person The only rule for a subjective probability is

that the probability of the event must be a value in the interval [0,1]

Subjective Probability Approach


Equally Likely (Laplace) The equally likely approach usually relies on

symmetry to assign probabilities to events◦ As such, previous research or experiments are not

needed to determine the probabilities Suppose that an experiment has only n outcomes

The equally likely approach to probability assigns a probability of 1/n to each of the outcomes

Further, if an event A is made up of m outcomes thenP(A) = m/n


Selecting a simple random sample of 2 individuals◦ Each pair has an equal probability of being

selected Rolling a fair die

◦ Probability of rolling a “4” is 1/6 This does not mean that whenever you roll the die 6

times, you always get exactly one “4”◦ Probability of rolling an even number

2,4, & 6 are all even so we have 3 possible outcomes in the event we want to examine

Thus the probability of rolling an even number is 3/6 = 1/2

Examples


Borrows from calculus’ concept of the limit

◦ We cannot repeat an experiment infinitely many times so instead we use a ‘large’ n Process

Repeat an experiment n times Record the number of times an event A occurs, denote

this value by a Calculate the value of a/n

Relative Frequency (von Mises)

17

n

aAP

n lim)(

n

aAP )(


“large” n?◦ Law of Large Numbers

As the number of repetitions of a random experiment increases, the chance that the relative frequency of occurrence for an event will differ from the true probability of the event by more than any small number approaches zero Doing a large number of repetitions allows us to

accurately approximate the true probabilities using the results of our repetitions

Relative Frequency Approach


Random Variables X is a random variable if the value that X

will assume cannot be predicted with certainty◦ That’s why its called random

Two types of random variables◦ Discrete

Can only assume a finite or countably infinite number of different values

◦ Continuous Can assume all the values in some interval


Examples Are the following random variables discrete

or continuous?◦ X = number of houses sold by a real estate

developer per week◦ X = weight of a child at birth◦ X = time required to run 800 meters◦ X = number of heads in ten tosses of a coin


Discrete Probability Distribution A list of the possible values of a random

variable X, say (xi) and the probability associated with each, P(X=xi)◦ All probabilities must be nonnegative◦ Probabilities sum to 1

21

0 ( ) 1

( ) 1i

i

P x

P x


Example

The table above gives the proportion of employees who use X number of sick days in a year◦ An employee is to be selected at random

Let X = # of days of leave P(X=2) = P(X≥4) = P(X<4) = P(1≤X≤6) =

22

X 0 1 2 3 4 5 6 7

P(X)

.1 .2 .2 .15 .1 .05 .05 .15


Expected Value of a Discrete Random Variable Expected Value (or mean) of a random

variable X◦ Mean = E(X) = μ = ΣxiP(X=xi)

Example

◦ E(X) =

23

X 2 4 6 8 10 12

P(X) .1 .05 .4 .25 .1 .1


Variance of a Discrete Random Variable Variance

◦ Var(X) = E(X-μ)2 = σ2 = Σ(xi-μ)2P(X=xi)

Example

◦ Var(X) =

24

X 2 4 6 8 10 12

P(X) .1 .05 .4 .25 .1 .1


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Lecture 6 Dustin Lueker. Standardized measure of variation ◦ Idea A standard deviation of 10 may indicate great variability or small variability,