DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 5 Discrete random variables and their probability.

DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS

Introduction to Business StatisticsQM 120

Chapter 5

Discrete random variables and their probability distribution

Chapter 5: Random Variables

Random variables

Let x denote the number of PCs owned by a family.

Then x can take any of the four possible values (0, 1, 2,

and 3).

A random variable (RV) is a variable whose value is

determined by the outcome of a random experiment.

2

# of PC’s owned Frequency Relative Frequency

0 120 .12

1 180 .18

2 470 .47

3 230 .23

N = 1000 Sum = 1.000

3


Discrete random variableA random variable that assumes countable values is called a discrete random variable.

Examples of discrete RVs:

Number of cars sold at a dealership during a week

Number of houses in a certain block

Number of fish caught on a fishing trip

Number of costumers in a bank at any given day

Continuous random variableA random variable that can assume any value contained in one or more intervals is called a continuous random variable.

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Examples of continuous RVs: Height of a person Time taken to complete a test Weight of a fish Price of a car

Example: Classify each of the following RVs as discrete or continuous.

The number of new accounts opened at a bank during a week

The time taken to run a marathon The price of a meal in fast food restaurant The score of a football game The weight of a parcel

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Chapter 5: Probability Distribution of a Discrete RV

The probability distribution of a discrete RV lists all the possible value that the RV can assume and their corresponding probabilities.

Example: Write the probability distribution of the number of PCs owned by a family.

# of PC’s owned

Frequency Relative Frequency

0 120 .12

1 180 .18

2 470 .47

3 230 .23

N = 1000 Sum = 1.000

X

0

1

2

3

P(x)

.12

.18

.47

.23

P(x) = 1.000

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The following two characteristics must hold for any discrete probability distribution:

The probability assigned to each value of a RV x lies in the range 0 to 1; that is 0 P(x) 1 for each x.

The sum of the probabilities assigned to all possible values of x is equal to 1.0; that is P(x) = 1.

Example: Each of the following tables lists certain values of x and their probabilities. Determine whether or not each table represents a valid probability distribution.


x P(x)

0 .08

1 .11

2 .39

3 .27

x P(x)

0 .25

1 .34

2 .28

3 .13

x P(x)

4 .2

5 .3

6 .6

8 -.1

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Example: The following table lists the probability

distribution of a discrete RV x.

a) P(x = 3) b) P(x 2) c) P(x 4) d) P(1 x 4)

e) Probability that x assumes a value less than 4

f) Probability that x assumes a value greater than 2

g) Probability that x assumes a value in the interval 2 to

5

x 0 1 2 3 4 5 6

P(x) .11 .19 .28 .15 .12 .09 .06


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Example: For the following table

a) Construct a probability distribution table. Draw a graph of the probability distribution.

b) Find the following probabilitiesi. P(x = 3) ii. P(x < 4) iii. P(x 3) iv. P(2 x 4)


x 1 2 3 4 5

f 8 20 24 16 12

x f

1 8

2 20

3 24

4 16

5 12

P(x)

.1

.25

.3

.2

.15

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Mean of a discrete RV

The mean -or expected value E(x)- of a discrete RV is the value that you would expect to observe on average if the experiment is repeated again and again

It is denoted by

Illustration: Let us toss two fair coins, and let x denote the number of heads observed. We should have the following probability distribution table

Suppose we repeat the experiment a large number of times, say n =4,000. We should expect to have approximately

( ) ( )E x xp x

x 0 1 2

P(x) 1/4 1/2 1/4

Chapter 5: Mean of a discrete RV

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1 thousand zeros, 2 thousand ones, and 1 thousand twos. Then the average value of x would equal

Similarly, if we use the , we would have

Sum of measurements 1,000

(0) 2,000(1) 1000(2)

4,000

1 1 1 (0) (1) (2)

4 4

2

n

( ) ( )E x xp x( ) 0 (0) 1 (1) 2 (2)

0(1/ 4) 1(1/ 2) 2(1/ 4)

1

E x P P P


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Example: Recall “ number of PC’s owned by a family” example. Find the mean number of PCs owned by a family.

Thus, on average, we expect to see 1.81 PC owned by a family!


x P(x)

0 .12

1 .18

2 .47

3 .23

We need to find x.p(x) for each value

of x and then add them up together

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Example: In a lottery conducted to benefit the local fire company, 8000 tickets are to be sold at $5 each. The prize is a $12,000. If you purchase two tickets, what is your expected gain?



Example: Determine the annual premium for a $1000 insurance policy covering an event that over a long period of time, has occurred at the rate of 2 times in 100. Let x : the yearly financial gain to the insurance company resulting from the sale of the policyC : unknown premiumCalculate the value of C such that the expected gain E(x) will equal to zero so that the company can add the administrative costs and profit.

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Example: You can insure a $50,000 diamond for its total value by paying a premium of D dollars. If the probability of theft in a given year is estimated to be .01, what premium should the insurance company charge if it wants the expected gain to equal $1000.

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Chapter 5: Standard Deviation of a Discrete RV

The standard deviation of a discrete RV x, denoted by , measures the spread of its probability distribution.

A higher value of indicates that x can assume values over a larger range about the mean. While, a smaller value indicates that most of the values that can x assume are clustered closely around the mean.

The standard deviation can be found using the following formula:

Hence, the variance 2 can be obtained by squaring its standard deviation .

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2 2( )x p x


Example: Recall “ number of PC’s owned by a family” example. Find the standard deviation of PCs owned by a family.

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Example: An electronic store sells a particular model of a computer notebook. There are only four notebooks in stocks, and the manager wonders what today’s demand for this particular model will be. She learns from marketing department that the probability distribution for x, the daily demand for the laptop, is as shown in the table.

Find the mean, variance, and the standard deviation of x. Is it likely that five or more costumers will want to buy a laptop?Solution:

17

x 0 1 2 3 4 5

P(x) .10 .40 .20 .15 .10 .05


Example: A farmer will earn a profit of $30,000 in case of heavy rain next year, $60,000 in case of moderate rain, and $15,000 in case of little rain. A meteorologist forecasts that the probability is .35 for heavy rain, .40 for moderate rain, and .25 for little rain next year. Let x be the RV that represents next year’s profits in thousands of dollars for this farmer. Write the probability distribution of x. Find the mean, variance, and the standard deviation of x.

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Example: An instant lottery ticket costs $2. Out of a total of 10,000 tickets printed for this lottery, 1000 tickets contain a prize of $5 each. 100 tickets contain a prize of $10 each, 5 tickets contain a prize of $1000 each, and 1 ticket has the prize of $5000. Let x be the RV that denotes the net amount player wins by playing this lottery. Write the probability distribution of x. Determine the mean and the standard deviation of x.

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Example: Based in its analysis of future demand for its products, the financial department a company has determined that there is a .17 probability that the company will lose $1.2 million during the next year, a .21 probability that it will lose $.7 million, a .37 probability that it will make a profit of $0.9 million, and a .25 probability that it will make a profit of $2.3 million.

a) Let x be a RV that denotes the profit earned by this company during the next year. Write the probability distribution of x.

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b) Find the mean and standard deviation of the probability of part a.

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Chapter 5: The Binomial Probability Distribution

The most widely used discrete probability

distribution.

It is applied to find the probability that an event will occur x times in n repetitions of an experiment (under certain conditions).

Suppose the probability that a VCR is defective at a factory is .05. We can apply the binomial probability distribution to find the odds of getting exactly one defective VCR out three VCRs.

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Binomial experimentAn experiment that satisfies the following four conditions is called a binomial experiment.

There are n identical trials Each trail has only two possible outcomes. The probabilities of the two outcomes remain

constant. The trials are independent.

A success does not mean that the corresponding outcome is considered favorable. Similarly, a failure doesn't necessarily refer to unfavorable outcome.

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Example: Consider the experiment consisting of 10 tosses of a coin. Determine whether or not it is a binomial experiment.

Solution:

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The binomial probability distribution and its formula.The RV x that represents the number of successes in n trials for a binomial experiment is called a binomial RV.

The probability distribution of x is called the binomial distribution.

Consider the VCRs example. Let x be the number of defective VCRs in a sample of 3, x can assume any of the values 0, 1, 2, 3.

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For a binomial experiment, the probability of exactly x successes in n trials is given by the binomial formula

where

n = total number of trials,

p = probability of success,

q = probability of failure 1 – p,

x = number of successes in n trials,

n – x = number of failures in n trials

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( )

!

!( )!

x n xn x

x n x

P x C p q

np q

x n x


Example: Find P(2) for a binomial RV with n = 5 and p =0.1

Solution:

27


Example: Over a long period of time it has been observed that a given sniper can hit a target on a single trial with a probability = .8. Suppose he fires four shots at the target.

Solution:

a) What is the probability that he will hit the target exactly two times?

b) What is the probability that he will hit the target at

least once?

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Example: Five percent of all VCRs manufactured by a large factory are defective. A quality control inspector selects three VCRs from the production line. What is the probability that exactly one of these three VCRs is defective

Solution:

29


Example: At the Express Delivery Services (EDS), providing high-quality service to its customers is the top priority of the management. The company guarantees a refund of all charges if a package is not delivered at its destination by the specified time. It is known from the past that despite all efforts, 2% of the packages mailed through this company does not arrive on time. A corporation mailed 10 packages through EDS.Solution:

a) Find the probability that exactly 1 of these 10 packages will not arrive on time

b) Find the probability that at most 1 of these 10 packages will not arrive on time.

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Example: According to the U.S. Bureau of Labor Statistics, 56% of mothers with children under 6 years of age work outside their homes. A random sample of 3 mothers with children under 6 years of age is selected. Let x denote the number of mothers who work outside their homes. Write the probability distribution of x and draw a graph of the probability distribution.

Solution:

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Using the table of binomial probabilitiesWe can use the tables of binomial probabilities (Table IV) given in Appendix C to calculate the required probabilities.

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P

n x .05 .1 .2 … .95

1 0 .9500 .9000 .8000 … .0500

1 .0500 .1000 .2000 … .9500

2 0 .9025 .8100 .6400 … .0025

1 .0950 .1800 .3200 … .0950

2 .0025 .0100 .0400 … .9025

3 0 .8574 .7290 .5120 … .0001

1 .1354 .2430 .3840 … .0071

2 .0071 .0270 .0960 … .1354

3 .0001 .0010 .0080 … .8574


Example: Based on data from A peter D. Hart Research Associates’ poll on consumer buying habits and attitudes, It was estimated that 5% of American shoppers are status shoppers, that is, shoppers who love to buy designer labels. A random sample of eight American shoppers is selected. Using Binomial Table, answer the following:Solution:

a) Find the probability that exactly 3 shoppers are status shoppers.

b) Find the probability that at most 2 shoppers are status shoppers.

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c) Find the probability that at least 3 shoppers are status shoppers.

d) Find the probability that 1 to 3 shoppers are status shoppers.

e) Let x be the number of status shoppers, Write the probability distribution of x and draw a graph of this probability.

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Probability of success and the shape of the

binomial distributionFor any number of trials n:

The binomial probability distribution is symmetric

if p = .5n

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p

n x .05 .1 … .5 … .9 .95

4 0 .8145 .6561 … .0625 … .0001 .0000

1 .1715 .2916 … .2500 … .0036 .0005

2 .0135 .0486 … .3750 … .0486 .0135

3 .0005 .0036 … .2500 … .2916 .1715

4 .0000 .0001 … .0625 … .6561 .81450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4

x

P(x)


The binomial probability distribution is skewed to the right if p is less than .5

36

p

n x .05 .1 … .5 … .9 .95

4 0 .8145 .6561 … .0625 … .0001 .0000

1 .1715 .2916 … .2500 … .0036 .0005

2 .0135 .0486 … .3750 … .0486 .0135

3 .0005 .0036 … .2500 … .2916 .1715

4 .0000 .0001 … .0625 … .6561 .8145

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4

x

P(x)


The binomial probability distribution is skewed to the left if p is greater than .5

37

p

n x .05 .1 … .5 … .9 .95

4 0 .8145 .6561 … .0625 … .0001 .0000

1 .1715 .2916 … .2500 … .0036 .0005

2 .0135 .0486 … .3750 … .0486 .0135

3 .0005 .0036 … .2500 … .2916 .1715

4 .0000 .0001 … .0625 … .6561 .8145

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4

x

P(x)


Mean and standard deviation of the binomial distributionAlthough we can still use the mean and standard deviations formulas learned in 5.3 and 5.4, it is more convenient and simpler to use the following formulas once the RV x is known to be a binomial RV

38

npqnp and


Example: Refer to the mothers with children under 6 years of age example, find the mean and the standard deviation of the probability distribution.

Solution:

39


Example: Let x be a discrete RV that posses a binomial distribution. Using the binomial formula, find the following probabilities.

a) P(x = 5) for n = 8 and p = .6

b) P(x = 3) for n = 4 and p = .3

c) P(x = 2) for n = 6 and p = .2

Verify your answer by using Table of Binomial Probabilities.

40


Example: Let x be a discrete RV that posses a binomial distribution.

a) Table of Binomial Probabilities, write the probability distribution for x for n = 7 and p = .3 and graph it.

b) What are the mean and the standard deviation of the probability distribution developed in part a?

41


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An experiment that satisfies the following four conditions is called a binomial experiment. There are n identical trials Each trail has only two possible outcomes. The probabilities of the two outcomes remain

constant. The trials are independent.

Chapter 5: The Hypergeometric Probability Distribution

We learned that one of the conditions required to apply the binomial distribution is that the trials are independent so the probability of the two outcomes remain constant.

What if the probability of the outcomes is not constant?

In such cases we replace the binomial distribution by the hypergeometric probability distribution.

Such a case occurs when a sample is drawn without replacement from a finite population.

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Hypergeometric probability distribution Let N = total number of elements in the

populationr = number of successes in the

populationN – r = number of failures in the

populationn = number of trials (sample size)x = number of successes in n trialsn – x = number of failures in n trials

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The mean and the variance are given by

The probability of x successes in n trials is given by


Example: Dawn corporation has 12 employees who hold managerial positions. Of them, 7 are female and 5 are male. The company is planning to send 3 of these 12 managers to a conference. If 3 mangers are randomly selected out of 12,

a) find the probability that all 3 of them are female

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b) find the probability that at most 1 of them is a female

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Example: A case of soda has 12 bottles, 3 of which contain diet soda. A sample of 4 bottles is randomly selected from the case

a) find the probability distribution of x, the number of diet sodas in the sample

47


b) what are the mean and variance of x?48


Example: GESCO Insurance company has prepared a final list of 8 candidates for 2 positions. Of the 8 candidates, 5 are business majors and 3 are engineers. If the company manager decides to select randomly two candidates from this list, find the probability that

a) both candidates are business majors

b) neither of the two candidates is a business major

c) at most one of the candidates is a business major

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Chapter 5: The Poisson Probability Distribution

The Poisson distribution is another discrete probability distribution that has numerous practical applications.

It provides a good model for data that represent the number of occurrence of a specified event in a given unit of time or space.

Here are some examples of experiments for which the RV x can modeled by the Poisson RV: The number of phone calls received by an operator

during a day The number of customer arrivals at checkout counter

during an hour The number of bacteria per a cm3 of a fluid The number of machine breakdowns during a given day The number of traffic accidents at a given time period

50


The Poisson probability distribution is applied to experiments with random and independent occurrences The occurrences are random in the sense they do not

follow any pattern and, hence, the are unpredictable. Independence of occurrences means that the

occurrence or nonoccurrence of an event does not influence the successive occurrence of that next event.

The occurrences are always considered with respect to an interval.

The interval may be a time interval, a space interval, or a volume interval.

If the average number of occurrences () for a given interval is known, then by using the Poisson probability we can compute the probability of a certain number of occurrences x in that interval.

51


The following three conditions must be satisfied to apply the Poisson probability distribution x is a discrete RV. The occurrences are random. The occurrences are independent.

The Poisson probability formula is given by

where (pronounced lambda) is the mean number of occurrences in that interval and the value of e is approximately 2.71828.

52

!)(

x

exP

x


The mean number of occurrences, denoted by , is called the parameter of the Poisson probability distribution or the Poisson parameter.

Remember: The interval of and x must be of the same length. If they are not, the mean should be redefined to make them equal. For instance, if was given in hours and we were asked to find the event in minutes, needs to be divided by 60 to have both x and in the same interval length

53


Example: The automatic teller machine (ATM) installed outside Mansfield Savings and Loan is used on average by five costumers per hour. The bank closed this ATM for one hour for repairs. What is the probability that during that hour eight customers came to use this ATM?

Solution:

54


Example: The average number of traffic accidents on a certain section of highway is two per week. assume that the number of accidents follows a Poisson distribution with = 2.

1. Find the probability of no accidents on this section of highway during a 1-week period.

2. Find the probability of at most three accidents on this section of highway during a 2-week period.

55


Using the table of Poisson probabilitiesThe probabilities for Poisson distribution can also be found using Poisson Probability Table

56


Example: On average, two new accounts are opened per day at an Imperial Savings Bank branches. Using Table VI of Appendix C, find the probability that on a given day the number of new accounts opened at this bank will be

a) Exactly 6 b) At most 3 c) At least 7

57


Mean and standard deviation of the Poisson probability distribution For the Poisson distribution

= 2 =

Example: An auto salesperson sells an average of .9 cars per day. Let x be the number of cars sold by this salesperson on any given day. Using the Poisson probability table, write the probability distribution of x, draw the probability distribution, and find the mean and standard deviation of x

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DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 5 Discrete random variables and their probability.

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