Concept explainers
A bookstore at the Hartsfield-Jackson Airport in Atlanta sells reading materials (paperback books, newspapers, magazines) as well as snacks (peanuts, pretzels, candy, etc.). A point-of-sale terminal collects a variety of information about customer purchases. Shown below is a table showing the number of snack items and the number of items of reading material purchased by the most recent 600 customers.
0 | Reading Material 1 |
2 | ||
0 | 0 | 60 | 18 | |
Snacks | 1 | 240 | 90 | 30 |
2 | 120 | 30 | 12 |
- a. Using the data in the table construct an empirical discrete bivariate
probability distribution for x = number of snack items and y = number of reading materials in a randomly selected customer purchase. what is the probability of a customer purchase consisting of one item of reading materials and two snack items? what is the probability of a customer purchasing one snack item only? why is the probability f(x = 0, y = 0) = 0? - b. Show the marginal probability distribution for the number of snack items purchased. Compute the
expected value and variance. - c. What is the expected value and variance for the number of reading materials purchased by a customer?
- d. Show the probability distribution for t = total number of items in a customer purchase. Compute its expected value and variance.
Compute the
a.
![Check Mark](/static/check-mark.png)
Develop an empirical bivariate probability distribution for number of snack items and the number of reading materials in a randomly selected customer purchase.
Find the probability of a customer purchase consisting of one item of reading materials and two snack items, probability of a customer purchasing one snack item and to explain why the probability,
Answer to Problem 62SE
The empirical bivariate probability distribution for number of snack items and the number of reading materials in a randomly selected customer purchase is as follows:
Reading material (y) | |||||
0 | 1 | 2 | Total | ||
Snacks | 0 | 0.00 | 0.10 | 0.03 | 0.13 |
1 | 0.40 | 0.15 | 0.05 | 0.60 | |
2 | 0.20 | 0.05 | 0.02 | 0.27 | |
Total | 0.60 | 0.30 | 0.10 | 1.00 |
The probability of a customer purchase consisting of one item of reading materials and two snack items is 0.05.
The probability of a customer purchasing one snack item is 0.40.
Explanation of Solution
Calculation:
The table represents the number of snack items and the number of items of reading material purchased by 600 customers. The random variable x represents the number of snack items and the random variable y represents the number of reading materials in a randomly selected customer purchase.
The bivariate distribution is obtained by dividing each frequency in the table by the total number of customers. The probability of a customer purchasing 1 item of reading material and 1 item of snacks is,
Similarly other bivariate probabilities are obtained and given in the following table:
Reading material (y) | |||||
0 | 1 | 2 | Total | ||
Snacks(x) | 0 | ||||
1 | |||||
2 | |||||
Total | 1.00 |
The marginal probability distribution for the random variable x is in the rightmost column and the marginal probability distribution for the random variable y is in the bottom row.
Thus, the bivariate probability distribution for number of snack items and the number of reading materials in a randomly selected customer purchase is obtained.
From the bivariate probability distribution table, the probability of a customer purchase consisting of one item of reading materials and two snack items is given by,
Thus, the probability of a customer purchase consisting of one item of reading materials and two snack items is 0.05.
From the bivariate probability distribution table, the probability of a customer purchase one snack item is given by,
Thus, the probability of a customer purchasing one snack item is 0.40.
The point sale terminal is only used when someone makes a purchase. Thus, the probability value,
b.
![Check Mark](/static/check-mark.png)
Show the marginal probability distribution for the number of snack items purchased and compute its expected value and variance.
Answer to Problem 62SE
The marginal probability distribution for the number of snack items purchased is:
x | f(x) |
0 | 0.13 |
1 | 0.60 |
2 | 0.27 |
The expected value for the number of snack items purchased is 1.14.
The variance for the number of snack items purchased is 0.3804.
Explanation of Solution
Calculation:
The marginal distribution for the number of snack items purchased is the values of the random variable x and the corresponding probabilities. This values are obtained from the rightmost column of the bivariate probability distribution for number of snack items and the number of reading materials in a randomly selected customer purchase obtained in part (a).
The marginal distribution of parts cost is given in the following table:
x | f(x) |
0 | 0.13 |
1 | 0.60 |
2 | 0.27 |
Thus, the marginal distribution of parts cost is obtained.
The expected value of the discrete random variable:
The formula for the expected value of a discrete random variable is,
The expected value for the random variable x is obtained in the following table:
x | f(x) | |
0 | 0.13 | 0.00 |
1 | 0.60 | 0.60 |
2 | 0.27 | 0.54 |
Total | 1.00 | 1.14 |
Thus, the expected value for the random variable x is 1.14.
The formula for the variance of the discrete random variable x is,
The variance of the random variable x is obtained using the following table:
x | f(x ) | |||
0 | 0.13 | –1.14 | 1.2996 | 0.16895 |
1 | 0.60 | –0.14 | 0.0196 | 0.01176 |
2 | 0.27 | 0.86 | 0.7396 | 0.19969 |
Total | 1.00 | –0.42 | 2.0588 | 0.3804 |
Therefore,
Thus, the variance of the random variable x is 0.3804.
c.
![Check Mark](/static/check-mark.png)
Compute the expected value and variance for the number of reading materials purchased by a customer.
Answer to Problem 62SE
The expected value for the number of reading materials purchased is 0.50.
The variance for the number of reading materials purchased is 0.45.
Explanation of Solution
Calculation:
The expected value of the discrete random variable:
The formula for the expected value of a discrete random variable is,
The expected value for the random variable y is obtained in the following table:
y | f(y) | |
0 | 0.60 | 0.0 |
1 | 0.30 | 0.30 |
2 | 0.10 | 0.2 |
Total | 1.0 | 0.5 |
Thus, the expected value for the random variable y is 0.5.
The formula for the variance of the discrete random variable y is,
The variance of the random variable y is obtained using the following table:
y | f(y) | |||
0 | 0.60 | –0.5 | 0.25 | 0.15 |
1 | 0.30 | 0.5 | 0.25 | 0.075 |
2 | 0.10 | 1.5 | 2.25 | 0.225 |
Total | 1.00 | 0.45 |
Therefore,
Thus, the variance of the random variable y is 0.45.
That is, the expected value for the number of reading materials purchased is 0.50 and the variance is 0.45.
d.
![Check Mark](/static/check-mark.png)
Show the probability distribution for total number of items in a customer purchase and to compute its expected value and variance.
Answer to Problem 62SE
The probability distribution for total number of items in a customer purchase is given below:
t | f(t) |
1 | 0.50 |
2 | 0.38 |
3 | 0.10 |
4 | 0.02 |
The expected number of items purchased is 1.64.
The variance for the total number of items purchased is 0.5504.
Explanation of Solution
Calculation:
The total number of items purchased is represented using the random variable t. the random variable t take the value 1 when
Substitute the values from the bivariate probability distribution table in part (a),
Similarly, other probability values can be obtained. Thus, the probability distribution for the random variable t is:
f(t) | |
1 | 0.50 |
2 | 0.38 |
3 | 0.10 |
4 | 0.02 |
e.
![Check Mark](/static/check-mark.png)
Compute the covariance and correlation coefficient between x and y.
Identify the relationship between the number of reading materials and the number of snacks purchased on a customer visit.
Answer to Problem 62SE
The covariance of x and y is –0.14 and the correlation coefficient is –0.3384.
There is a negative correlation between the number of reading materials and the number of snacks purchased on a customer visit.
Explanation of Solution
Calculation:
Covariance for the random variables x and y is given by,
Substitute the values,
Correlation between the random variables x and y is given by,
The standard deviation is obtained by taking the square root of variance.
Substitute the values in the correlation formula,
The covariance of x and y is –0.14 and the correlation coefficient is –0.3384.
Thus, correlation coefficient of x and y is –0.3384 which is negative. Thus, there is a negative correlation between the number of reading materials and the number of snacks purchased on a customer visit. That is, as the purchase of reading materials increase the number of snack items purchased decreases.
Want to see more full solutions like this?
Chapter 5 Solutions
Essentials Of Statistics For Business & Economics
- Examine the Variables: Carefully review and note the names of all variables in the dataset. Examples of these variables include: Mileage (mpg) Number of Cylinders (cyl) Displacement (disp) Horsepower (hp) Research: Google to understand these variables. Statistical Analysis: Select mpg variable, and perform the following statistical tests. Once you are done with these tests using mpg variable, repeat the same with hp Mean Median First Quartile (Q1) Second Quartile (Q2) Third Quartile (Q3) Fourth Quartile (Q4) 10th Percentile 70th Percentile Skewness Kurtosis Document Your Results: In RStudio: Before running each statistical test, provide a heading in the format shown at the bottom. “# Mean of mileage – Your name’s command” In Microsoft Word: Once you've completed all tests, take a screenshot of your results in RStudio and paste it into a Microsoft Word document. Make sure that snapshots are very clear. You will need multiple snapshots. Also transfer these results to the…arrow_forward2 (VaR and ES) Suppose X1 are independent. Prove that ~ Unif[-0.5, 0.5] and X2 VaRa (X1X2) < VaRa(X1) + VaRa (X2). ~ Unif[-0.5, 0.5]arrow_forward8 (Correlation and Diversification) Assume we have two stocks, A and B, show that a particular combination of the two stocks produce a risk-free portfolio when the correlation between the return of A and B is -1.arrow_forward
- 9 (Portfolio allocation) Suppose R₁ and R2 are returns of 2 assets and with expected return and variance respectively r₁ and 72 and variance-covariance σ2, 0%½ and σ12. Find −∞ ≤ w ≤ ∞ such that the portfolio wR₁ + (1 - w) R₂ has the smallest risk.arrow_forward7 (Multivariate random variable) Suppose X, €1, €2, €3 are IID N(0, 1) and Y2 Y₁ = 0.2 0.8X + €1, Y₂ = 0.3 +0.7X+ €2, Y3 = 0.2 + 0.9X + €3. = (In models like this, X is called the common factors of Y₁, Y₂, Y3.) Y = (Y1, Y2, Y3). (a) Find E(Y) and cov(Y). (b) What can you observe from cov(Y). Writearrow_forward1 (VaR and ES) Suppose X ~ f(x) with 1+x, if 0> x > −1 f(x) = 1−x if 1 x > 0 Find VaRo.05 (X) and ES0.05 (X).arrow_forward
- Joy is making Christmas gifts. She has 6 1/12 feet of yarn and will need 4 1/4 to complete our project. How much yarn will she have left over compute this solution in two different ways arrow_forwardSolve for X. Explain each step. 2^2x • 2^-4=8arrow_forwardOne hundred people were surveyed, and one question pertained to their educational background. The results of this question and their genders are given in the following table. Female (F) Male (F′) Total College degree (D) 30 20 50 No college degree (D′) 30 20 50 Total 60 40 100 If a person is selected at random from those surveyed, find the probability of each of the following events.1. The person is female or has a college degree. Answer: equation editor Equation Editor 2. The person is male or does not have a college degree. Answer: equation editor Equation Editor 3. The person is female or does not have a college degree.arrow_forward
- Algebra: Structure And Method, Book 1AlgebraISBN:9780395977224Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. ColePublisher:McDougal LittellMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Holt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGAL
- College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage LearningElementary Geometry for College StudentsGeometryISBN:9781285195698Author:Daniel C. Alexander, Geralyn M. KoeberleinPublisher:Cengage Learning
![Text book image](https://www.bartleby.com/isbn_cover_images/9780395977224/9780395977224_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
![Text book image](https://www.bartleby.com/isbn_cover_images/9780547587776/9780547587776_smallCoverImage.jpg)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781285195698/9781285195698_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781337282291/9781337282291_smallCoverImage.gif)