x₂ X₂ 0 1 2 3 0.09 0.06 0.04 0.00 1 0.05 0.15 0.05 0.04 2 0.05 0.04 0.10 0.06 3 0.00 0.03 0.04 0.07 40.00 0.01 0.05 0.06 (a) What is P(x,-1, x₂-1), that is, the probability that there is exactly one customer in each line? PCX₂-1, x₂-1)-015✔ (b) What is P(x,-X₂), that is, the probability that the numbers of customers in the two lines are identical? PCX₂-x₂) x (c) Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X, and X₂. OA-(x, s2+x₂ UX₂ z 2 + x₂) OA= {X₂ ≥2+ X₂ UX₂ ≥2+x₂) OA (X₁ ≤2+x₂ UX₂ ≤2+x₂) OA-(X, 22+x₂ UX₂ ≤2+x₂) Calculate the probability of this event. P(A)- x x (d) What is the probability that the total number of customers in the two lines is exactly four? At least four? P(exactly four)- x P(at least four)-1
x₂ X₂ 0 1 2 3 0.09 0.06 0.04 0.00 1 0.05 0.15 0.05 0.04 2 0.05 0.04 0.10 0.06 3 0.00 0.03 0.04 0.07 40.00 0.01 0.05 0.06 (a) What is P(x,-1, x₂-1), that is, the probability that there is exactly one customer in each line? PCX₂-1, x₂-1)-015✔ (b) What is P(x,-X₂), that is, the probability that the numbers of customers in the two lines are identical? PCX₂-x₂) x (c) Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X, and X₂. OA-(x, s2+x₂ UX₂ z 2 + x₂) OA= {X₂ ≥2+ X₂ UX₂ ≥2+x₂) OA (X₁ ≤2+x₂ UX₂ ≤2+x₂) OA-(X, 22+x₂ UX₂ ≤2+x₂) Calculate the probability of this event. P(A)- x x (d) What is the probability that the total number of customers in the two lines is exactly four? At least four? P(exactly four)- x P(at least four)-1
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Educational Website Content
#### Joint Probability Mass Function (pmf) Analysis
A certain market has both an express checkout line and a superexpress checkout line. Let \( X_1 \) denote the number of customers in line at the express checkout at a particular time of day, and let \( X_2 \) denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of \( X_1 \) and \( X_2 \) is given in the accompanying table.
#### Joint PMF of \( X_1 \) and \( X_2 \)
| \( X_1 \backslash X_2 \) | 0 | 1 | 2 | 3 |
|--------------------------|-------|-------|-------|-------|
| 0 | 0.09 | 0.02 | 0.03 | 0.1 |
| 1 | 0.09 | 0.05 | 0.05 | 0.04 |
| 2 | 0.06 | 0.1 | 0.08 | 0.04 |
| 3 | 0.07 | 0.05 | 0.04 | 0.06 |
1. **What is \( P(X_1 = 1, X_2 = 1) \)**, that is, the probability that there is exactly one customer in each line?
\[
P(X_1 = 1, X_2 = 1) = 0.05
\]
2. **What is \( P(X_1 = X_2) \)**, that is, the probability that the numbers of customers in the two lines are identical?
\[
P(X_1 = X_2) = \sum_{i=0}^{3} P(X_1 = i, X_2 = i) = 0.09 + 0.05 + 0.08 + 0.06 = 0.28
\]
3. Let \( A \) denote the event that there are at least two more customers in one line than in the other line. Express \( A \) in terms of \( X_1 \) and \( X_2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feeb0cd63-fcaf-4f78-a653-01ec8225914a%2F6503a671-7287-4ff9-b187-49f31527d32c%2Fh9afhs_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Website Content
#### Joint Probability Mass Function (pmf) Analysis
A certain market has both an express checkout line and a superexpress checkout line. Let \( X_1 \) denote the number of customers in line at the express checkout at a particular time of day, and let \( X_2 \) denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of \( X_1 \) and \( X_2 \) is given in the accompanying table.
#### Joint PMF of \( X_1 \) and \( X_2 \)
| \( X_1 \backslash X_2 \) | 0 | 1 | 2 | 3 |
|--------------------------|-------|-------|-------|-------|
| 0 | 0.09 | 0.02 | 0.03 | 0.1 |
| 1 | 0.09 | 0.05 | 0.05 | 0.04 |
| 2 | 0.06 | 0.1 | 0.08 | 0.04 |
| 3 | 0.07 | 0.05 | 0.04 | 0.06 |
1. **What is \( P(X_1 = 1, X_2 = 1) \)**, that is, the probability that there is exactly one customer in each line?
\[
P(X_1 = 1, X_2 = 1) = 0.05
\]
2. **What is \( P(X_1 = X_2) \)**, that is, the probability that the numbers of customers in the two lines are identical?
\[
P(X_1 = X_2) = \sum_{i=0}^{3} P(X_1 = i, X_2 = i) = 0.09 + 0.05 + 0.08 + 0.06 = 0.28
\]
3. Let \( A \) denote the event that there are at least two more customers in one line than in the other line. Express \( A \) in terms of \( X_1 \) and \( X_2 \).
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