Chapter 6
Business Statistics I
Dr. Changping Wang
Chapter 6 – Discrete Probability Distributions 6.1. Discrete Probability Distributions
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Business Statistics I
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Dr. Changping Wang
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Business Statistics I
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Dr. Changping Wang
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Business Statistics I
Dr. Changping Wang
Example 1: Toss a fair coin three times. Let X be the number of heads. Write down the probability distribution of X.
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Business Statistics I
Dr. Changping Wang
Example 2: At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy one ticket. Let X be the value of your gain. Write down the probability distribution of X.
Session 7
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Business Statistics I
Dr. Changping Wang
Mean (or Expected Value) of a (discrete) random variable Given a discrete probability distribution X x1 x 2 … x … P ( X = x ) p1 p 2 ... p … n
i
n
Then the mean, denoted by E(X), is xi p i E ( X ) = ∑ i
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Business Statistics I
Dr. Changping Wang
Example 3: Roll a fair die. Let X be the number rolled. Then X P(X) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 Find the mean E(X).
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Business Statistics I
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Dr. Changping Wang
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Business Statistics I
Dr. Changping Wang
Example 4: Roll a fair die. Let X be the number rolled. Then X P(X) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 Find the variance σ 2 (or Var(X)) and standard deviation σ of X.
Note: We are not interested in showing the use of the formula, since you can use a CASIO calculator to compute them. Session 7
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Business Statistics I
Dr. Changping Wang
For instance, we can put the distribution in a calculator, and treat it as a weighted data. Then we can find out mean and standard deviation.
LIST 1 LIST 2 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 XLIST: LIST 1 Freq: LIST 2
Expected value Decision Making Example 5.7 (Page 232 on the text) Session 7
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Business Statistics I
Dr. Changping Wang
A fast food company plans to install a new icecream dispensing unit in one of two store locations. The company figures that the probability of a unit being successful in location A is ¾ and the annual profit in this case is $150,000. If it is not successful, there will be losses of $80,000. At location B the probability of succeeding is ½, and the potential profit and loss are $240,000 and $48,000, respectively. a) Where should the company locate to maximize the expected profit?
b) Which location is less risky, i.e., has the lower relative variability?
Example 7: B. F. Retread, a tire manufacturer, wants to select one of the feasible designs for a new longer wearing radial tire. The manufacturing cost of each type of tire is shown below. Tire Design
Session 7
Fixed Cost per year
Variable Cost per tire Page 11
Business Statistics I
Dr. Changping Wang
A
$60,000
$30
B
$90,000
$20
C
$120,000
$15
There are 3 possible levels of annual demand: 5,000 tires, 7,000 tires and 11,000 tires. The respective probabilities are 0.2, 0.4 and 0.4. The selling prices for A, B and C will be $85, $65 and $75, respectively. Question 1 Based on expected profit, which design should be produced? (a) C (b) B (c) A (d) A or B (e) B or C
Question 2 What is the expected profit for the design B? a. $372,000 b. $279,000 c. $313,500 d. $391,000 e. None of these
Example 8: If we roll a fair die 3 times, find out the probability that we will roll a number “1” exactly 2 times. Solution. If we use a triple (a, b, c) to denote an outcome of such experiment, then it means that we roll a die three times, and we roll numbers a, b and c at the first, second and third times, respectively. So, the sample space is S={(1,1,1), (1,1,2), (1,1,3), (1,1,4), (1,1,5), (1,1,6), (1,2,1), (1,2,2), (1,2,3), (1,2,4), (1,2,5), (1,2,6), Session 7
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Business Statistics I
Dr. Changping Wang
(1,3,1), (1,3,2), (1,3,3), (1,3,4), (1,3,5), (1,3,6), (1,4,1), (1,4,2), (1,4,3), (1,4,4), (1,4,5), (1,4,6), (1,5,1), (1,5,2), (1,5,3), (1,5,4), (1,5,5), (1,5,6), (1,6,1), (1,6,2), (1,6,3), (1,6,4), (1,6,5), (1,6,6), (2,1,1), (2,1,2), (2,1,3), (2,1,4), (2,1,5), (2,1,6), (2,2,1), (2,2,2), (2,2,3), (2,2,4), (2,2,5), (2,2,6), ……… (2,6,1), (2,6,2), (2,6,3), (2,6,4), (2,6,5), (2,6,6), (3,1,1), (3,1,2), (3,1,3), (3,1,4), (3,1,5), (3,1,6), ……… (3,6,1), (3,6,2), (3,6,3), (3,6,4), (3,6,5), (3,6,6), (4,1,1), (4,1,2), (4,1,3), (4,1,4), (4,1,5), (4,1,6), ……… (4,6,1), (4,6,2), (4,6,3), (4,6,4), (4,6,5), (4,6,6), (5,1,1), (5,1,2), (5,1,3), (5,1,4), (5,1,5), (5,1,6), ……… (5,6,1), (5,6,2), (5,6,3), (5,6,4), (5,6,5), (5,6,6), (6,1,1), (6,1,2), (6,1,3), (6,1,4), (6,1,5), (6,1,6), ……… (6,6,1), (6,6,2), (6,6,3), (6,6,4), (6,6,5), (6,6,6)} So, the probability that we will roll a number 2 exactly 2 times is 15/216=5/72 6.2. Binomial Distributions James Bernoulli (Swiss Mathematician)
Session 7
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Business Statistics I
Dr. Changping Wang
NOTE: In the text, π is used instead of p. Session 7
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Business Statistics I
Dr. Changping Wang
Example 9. Twentysix percent of couples who plan to marry this year are planning destination weddings. In a random sample of 12 couples who plan to marry, find the probability that a. Exactly 6 couples will have a destination wedding.
b. At least 6 couples will have a destination wedding.
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Business Statistics I
Dr. Changping Wang
c. Fewer than 5 couples will have a destination wedding.
Calculator Lesson 3 (Page 282): How to use Calculator to compute the binomial probabilities? Step 1: Use STAT mode Step 2: Use the F6 key to get the following menu [GRPH ] [CALC ] [TEST ] [ INTR ] [ DIST ] [>>] F1 F2 F3 F4 F5 F6 Step 3: Use the F5 key to get the following menu [ NORM ] [t ] [CHI ] [ F ] [ BINM ] [>>] F1 F2 F3 F4 F5 F6 Step 4: Use the F5 key to get the following menu [ Bpd ] [ Bcd ] [ InvB] F1 F2 F3 Step 5: Use the F1 or F2 to get the following display [ Bpd ] [ Bcd ] [ InvB ] F1 F2 F3 Note: Bpd(k, n, p)=P(X=k) Bcd(k, n,p)=P(X
Dr. Changping Wang
Chapter 6 – Discrete Probability Distributions 6.1. Discrete Probability Distributions
Session 7
Page 1
Business Statistics I
Session 7
Dr. Changping Wang
Page 2
Business Statistics I
Session 7
Dr. Changping Wang
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Business Statistics I
Dr. Changping Wang
Example 1: Toss a fair coin three times. Let X be the number of heads. Write down the probability distribution of X.
Session 7
Page 4
Business Statistics I
Dr. Changping Wang
Example 2: At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy one ticket. Let X be the value of your gain. Write down the probability distribution of X.
Session 7
Page 5
Business Statistics I
Dr. Changping Wang
Mean (or Expected Value) of a (discrete) random variable Given a discrete probability distribution X x1 x 2 … x … P ( X = x ) p1 p 2 ... p … n
i
n
Then the mean, denoted by E(X), is xi p i E ( X ) = ∑ i
Session 7
Page 6
Business Statistics I
Dr. Changping Wang
Example 3: Roll a fair die. Let X be the number rolled. Then X P(X) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 Find the mean E(X).
Session 7
Page 7
Business Statistics I
Session 7
Dr. Changping Wang
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Business Statistics I
Dr. Changping Wang
Example 4: Roll a fair die. Let X be the number rolled. Then X P(X) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 Find the variance σ 2 (or Var(X)) and standard deviation σ of X.
Note: We are not interested in showing the use of the formula, since you can use a CASIO calculator to compute them. Session 7
Page 9
Business Statistics I
Dr. Changping Wang
For instance, we can put the distribution in a calculator, and treat it as a weighted data. Then we can find out mean and standard deviation.
LIST 1 LIST 2 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 XLIST: LIST 1 Freq: LIST 2
Expected value Decision Making Example 5.7 (Page 232 on the text) Session 7
Page 10
Business Statistics I
Dr. Changping Wang
A fast food company plans to install a new icecream dispensing unit in one of two store locations. The company figures that the probability of a unit being successful in location A is ¾ and the annual profit in this case is $150,000. If it is not successful, there will be losses of $80,000. At location B the probability of succeeding is ½, and the potential profit and loss are $240,000 and $48,000, respectively. a) Where should the company locate to maximize the expected profit?
b) Which location is less risky, i.e., has the lower relative variability?
Example 7: B. F. Retread, a tire manufacturer, wants to select one of the feasible designs for a new longer wearing radial tire. The manufacturing cost of each type of tire is shown below. Tire Design
Session 7
Fixed Cost per year
Variable Cost per tire Page 11
Business Statistics I
Dr. Changping Wang
A
$60,000
$30
B
$90,000
$20
C
$120,000
$15
There are 3 possible levels of annual demand: 5,000 tires, 7,000 tires and 11,000 tires. The respective probabilities are 0.2, 0.4 and 0.4. The selling prices for A, B and C will be $85, $65 and $75, respectively. Question 1 Based on expected profit, which design should be produced? (a) C (b) B (c) A (d) A or B (e) B or C
Question 2 What is the expected profit for the design B? a. $372,000 b. $279,000 c. $313,500 d. $391,000 e. None of these
Example 8: If we roll a fair die 3 times, find out the probability that we will roll a number “1” exactly 2 times. Solution. If we use a triple (a, b, c) to denote an outcome of such experiment, then it means that we roll a die three times, and we roll numbers a, b and c at the first, second and third times, respectively. So, the sample space is S={(1,1,1), (1,1,2), (1,1,3), (1,1,4), (1,1,5), (1,1,6), (1,2,1), (1,2,2), (1,2,3), (1,2,4), (1,2,5), (1,2,6), Session 7
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Business Statistics I
Dr. Changping Wang
(1,3,1), (1,3,2), (1,3,3), (1,3,4), (1,3,5), (1,3,6), (1,4,1), (1,4,2), (1,4,3), (1,4,4), (1,4,5), (1,4,6), (1,5,1), (1,5,2), (1,5,3), (1,5,4), (1,5,5), (1,5,6), (1,6,1), (1,6,2), (1,6,3), (1,6,4), (1,6,5), (1,6,6), (2,1,1), (2,1,2), (2,1,3), (2,1,4), (2,1,5), (2,1,6), (2,2,1), (2,2,2), (2,2,3), (2,2,4), (2,2,5), (2,2,6), ……… (2,6,1), (2,6,2), (2,6,3), (2,6,4), (2,6,5), (2,6,6), (3,1,1), (3,1,2), (3,1,3), (3,1,4), (3,1,5), (3,1,6), ……… (3,6,1), (3,6,2), (3,6,3), (3,6,4), (3,6,5), (3,6,6), (4,1,1), (4,1,2), (4,1,3), (4,1,4), (4,1,5), (4,1,6), ……… (4,6,1), (4,6,2), (4,6,3), (4,6,4), (4,6,5), (4,6,6), (5,1,1), (5,1,2), (5,1,3), (5,1,4), (5,1,5), (5,1,6), ……… (5,6,1), (5,6,2), (5,6,3), (5,6,4), (5,6,5), (5,6,6), (6,1,1), (6,1,2), (6,1,3), (6,1,4), (6,1,5), (6,1,6), ……… (6,6,1), (6,6,2), (6,6,3), (6,6,4), (6,6,5), (6,6,6)} So, the probability that we will roll a number 2 exactly 2 times is 15/216=5/72 6.2. Binomial Distributions James Bernoulli (Swiss Mathematician)
Session 7
Page 13
Business Statistics I
Dr. Changping Wang
NOTE: In the text, π is used instead of p. Session 7
Page 14
Business Statistics I
Dr. Changping Wang
Example 9. Twentysix percent of couples who plan to marry this year are planning destination weddings. In a random sample of 12 couples who plan to marry, find the probability that a. Exactly 6 couples will have a destination wedding.
b. At least 6 couples will have a destination wedding.
Session 7
Page 15
Business Statistics I
Dr. Changping Wang
c. Fewer than 5 couples will have a destination wedding.
Calculator Lesson 3 (Page 282): How to use Calculator to compute the binomial probabilities? Step 1: Use STAT mode Step 2: Use the F6 key to get the following menu [GRPH ] [CALC ] [TEST ] [ INTR ] [ DIST ] [>>] F1 F2 F3 F4 F5 F6 Step 3: Use the F5 key to get the following menu [ NORM ] [t ] [CHI ] [ F ] [ BINM ] [>>] F1 F2 F3 F4 F5 F6 Step 4: Use the F5 key to get the following menu [ Bpd ] [ Bcd ] [ InvB] F1 F2 F3 Step 5: Use the F1 or F2 to get the following display [ Bpd ] [ Bcd ] [ InvB ] F1 F2 F3 Note: Bpd(k, n, p)=P(X=k) Bcd(k, n,p)=P(X
