Statistical techniques in business ecohomics chap006

Chapter SixDiscrete Probability Distributions GOALS When you have completed this chapter, you will be able to: ONE Define the terms random variable and probability... Chapter Six contin

Chapter Six

Discrete Probability Distributions

GOALS

When you have completed this chapter, you will

be able to: ONE

Define the terms random variable and probability

Chapter Six continued

Discrete Probability Distributions

GOALS

When you have completed this chapter, you will

be able to:

FOUR

Describe the characteristics and compute probabilities

using the binomial probability distribution

FIVE

Describe the characteristics and compute probabilities

using the hypergeometric distribution

SIX

Describe the characteristics and compute the

probabilities using the Poisson distribution

Types of Probability Distributions

Discrete probability Distribution

Can assume only certain

outcomes

Continuous Probability Distribution

Can assume an infinite number of

values within a given range

Continuous Probability Distribution

Features of a Discrete Distribution

Discrete Probability Distribution

The sum of the

1.00.

The outcomes are mutually exclusive.

The number of students in a class

The number of children in a family

The number of

cars entering a

carwash in a hour

Thus the possible

coin is tossed three

times Let x be the

number of heads Let

H represent the

outcome of a head and

T the outcome of a tail.

EXAMPLE 1 continued

The outcome of zero heads occurred once.

The outcome

of one head occurred three times.

outcome of two heads occurred three times.

The Mean of a Discrete Probability Distribution

)]

( [ xP x







Mean

The central location of the data

The long-run average value

of the random variable

Also referred to as its

expected value, E(X), in a

The Variance of a Discrete Probability Distribution

Painted weeks of weeks

Dan Desch, owner of College Painters, studied his

records for the past 20 weeks and reports the

following number of houses painted per week.

P hysics

EXAMPLE 2

)] (

per week

) [( 2

   

Variance in the number of

houses painted per week

Binomial Probability

Distribution

The trials are independent.

Binomial Probability Distribution

counts.

The probability

of success stays the same for each trial.

n is the number of trials

x is the number of observed successes

p is the probability of success on each trial

x n

x x

n C x

P ( )   ( 1   ) 

Binomial Probability Distribution

000

172

250

)80(

)20(

)80(

)20(

)3

The Alabama Department

of Labor reports that 20%

of the workforce in Mobile

is unemployed and

interviewed 14 workers.

What is the probability that exactly three are unemployed?

At least three are

unemployed?

2501

) 0859 )(.

0080 )(.

364 (

) 20 1 ( ) 20 (.

) 3 ( 14 3 3 11

Example 3

956

044

1

) 20

1 ( ) 20 (.

1

) 0 ( 1

) 1

(

14

00

P

The probability at least one is unemployed?

Mean & Variance of the Binomial

Distribution

 n 

 2  n (  1   )

Mean of the Binomial Distribution

Variance of the Binomial Distribution

Example 3 Revisited

Mean and Variance Example

The number of cars

in the parking lot

Hypergeometric Distribution

Hypergeometric Distribution

Only 2 possible outcomes

Sampling from a finite population without replacement, the probability of

a success is not the same on each trial.

Results from a count of the number of successes in a fixed number of trials Trials are not independent

N is the size of the population

S is the number of successes in the population

x is the number of successes in a sample of n

observations

Hypergeometric Distribution

Hypergeometric Distribution

The size of the sample n is

greater than 5% of the

size of the population N

The sample is selected from a finite population without replacement (recall that a criteria for the binomial distribution is that the probability

of success remains the same from trial to trial).

Use to find the probability

of a specified number of

successes or failures if

EXAMPLE 5

What is the probability that three of five

violations randomly selected to be investigated are actually violations?

are actual violations

and the Security

Agency will only be

able to investigate

five of the violations.

The limiting form of the

binomial distribution

where the probability of

probability distribution

The binomial distribution becomes more skewed to the right (positive) as the probability of success

become smaller.

238

252

) 15 (

4 )

)(

(

)(

( )

3 (

5 10

2 6

3 4

5 10

2 5 4

10 3

C

C

C P

Poisson probability distribution

Poisson probability distribution

 is the mean number of successes

in a particular interval of time

Variance

Also equal to np

Poisson Probability Distribution

EXAMPLE 6

1465

.

! 2

4

!

) (

P

u x



The Sylvania Urgent

Care facility specializes

in caring for minor

injuries, colds, and flu

For the evening hours of