Business Statistics 4

7/29/2019 Business Statistics 4

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Discrete Probability Distributions

Chapter-4


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Probability Distribution

Probabilities tell you???

how likely certain events.

What probability does not tell.

overall impact of these events

and, what it means to you?

In this chapter we will see how to use

probability to predict long-term outcomes and

also measure the certainty of these predictions


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Probability Distribution

Random Variables

Discrete Probability Distributions

Expected value and Variance Binomial Distribution

Poisson Distribution

Hypergeometric Distribution


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Random Variables

A random variable is a numerical description of theoutcome of an experiment.

A discrete random variable may assume either a

finite number of values or an infinite sequence ofvalues.

A continuous random variable may assume any

numerical value in an interval or collection ofintervals. Eg:- Time, weight, distance, and temperaturecan be described by continuous random variables.

Which is a variable that

can take a set of values,

where each value is

associated with a

specific probability


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Example: XYZ MartNumber of customers

visiting the store

No. of days this level was

Observed

Probability that Random

Variable will take on this

value

101 5 0.05

102 7 0.07

1038 0.08

104 9 0.09

105 10 0.1

106 12 0.12

107 12 0.12

108 11 0.11

109 10 0.1

110 9 0.09

111 7 0.08

Total 100 1.00


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Example: XYZ Mart

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

101 102 103 104 105 106 107 108 109 110 111

Probability

Probability


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Random Variables

Question Random Variable x Type

Family

size

x = Number of dependents

reported on tax return

Discrete

Distance from

home to store

x = Distance in miles from

home to the store site

Continuous

Own dogor cat

x = 1 if own no pet;= 2 if own dog(s) only;

= 3 if own cat(s) only;

= 4 if own dog(s) and cat(s)

Discrete


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Discrete Probability Distribution

The probability distribution for a random variabledescribes how probabilities are distributed overthe values of the random variable.

We can describe a discrete probability distributionwith a table, graph, or equation.


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The probability distribution is defined by aprobability function, denoted byf(x), which providesthe probability for each value of the random variable.

The required conditions for a discrete probabilityfunction are:

f(x) > 0

f(x) = 1


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A tabular representation of the probability distribution for TV sale

Number

Units Sold of Days0 80

1 50

2 40

3 104 20

200

x f(x)0 .40

1 .25

2 .20

3 .054 .10

1.00

80/200


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.10

.20

.30

.40

.50

0 1 2 3 4Values of Random Variable x (TV sales)

Probabi

lity

Graphical Representation of Probability Distribution


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Discrete Uniform Probability

Distribution

The discrete uniform probability distribution is thesimplest example of a discrete probabilitydistribution given by a formula.

The discrete uniform probability function is

f(x) = 1/n

where:n = the number of values the random

variable may assume

the values of therandom variable

are equally likely


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Expected Value

The expected value, or mean, of a random variableis a measure of its central location.

The variance summarizes the variability in thevalues of a random variable.

The standard deviation, , is defined as the positivesquare root of the variance.

Var(x) = 2 = (x - )2f(x)

E(x) = = xf(x)


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Expected Value

expected number ofTVs sold in a day

x f(x) xf(x)

0 .40 .00

1 .25 .25

2 .20 .403 .05 .15

4 .10 .40

E(x) = 1.20


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Expected Value

Variance and Standard Deviation

0

12

3

4

-1.2

-0.20.8

1.8

2.8

1.44

0.040.64

3.24

7.84

.40

.25

.20

.05

.10

.576

.010

.128

.162

.784

x - (x - )2 f(x) (x - )2f(x)

Variance of daily sales = 2 = 1.660

x

Standard deviation of daily sales = 1.2884 TVs


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Binomial Distribution

It is associated with a multi-step experiment

Four Properties of a Binomial Experiment

3. The probability of a success, denoted byp, doesnot change from trial to trial.

4. The trials are independent.

2. Two outcomes, success and failure, are possibleon each trial.

1. The experiment consists of a sequence of nidentical trials.


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where:

f(x) = the probability of x successes in n trials

n = the number of trialsp = the probability of success on any one trial

( )!( ) (1 )!( )!

x n xnf x p px n x

Binomial Probability Function

Could x be a Continuous Random Variable?


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Example:-

Evans is concerned about a low retention rate foremployees. In recent years, management has seen aturnover of 10% of the hourly employees annually.Thus, for any hourly employee chosen at random,

management estimates a probability of 0.1 that theperson will not be with the company next year.


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Using the Binomial Probability Function

Choosing 3 hourly employees at random, what isthe probability that 1 of them will leave the companythis year?

f xn

x n xp px n x( )

!

!( )!( )

( )

1

1 23!(1) (0.1) (0.9) 3(.1)(.81) .2431!(3 1)!

f

Let: p = .10, n = 3, x = 1


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DecisionTree


1st Worker 2nd Worker 3rd Worker x Prob.

Leaves

(.1)

Stays(.9)

3

2

0

2

2

Leaves (.1)

Leaves (.1)

S (.9)

Stays (.9)

Stays (.9)

S (.9)

S (.9)

S (.9)

L (.1)

L (.1)

L (.1)

L (.1) .0010

.0090

.0090

.7290

.0090

1

1

.0810

.0810

.0810

1


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(1 )np p

E(x) = = np

Var(x) = 2 = np(1 p)

Expected Value

Variance

Standard Deviation


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3(.1)(.9) .52 employees

E(x) = = 3(.1) = .3 employees out of 3

Var(x) = 2 = 3(.1)(.9) = .27

Expected Value

Variance

Standard Deviation


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Binomial DistributionHarish is in charge of the electronics section of a large

departmental store. He has noticed that the probability

that a customer who is just browsing will buy somethingis 0.3. Suppose that 15 customers browse in the

electronics section each Hour, then

1. What is the Prob. that at least one browsing customerwill buy something during a specific hour.

2. What is the Prob. that at least four browsing customer

will buy something during a specific hour.

3. What is the Prob. that no browsing customer will buy

anything during a specific hour.

4. What is the Prob. that no more than four browsing

customers will buy something during a specific hour.

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