There are two traffic lights on a commuter's route to and from work. Let X, be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning from work. Suppose that these two variables are independent, each with the pmf given in the accompanying table (so X, X2 is a random sample of size n- 2). x o 1 2 P(x)0.3 0.4 0.3 -1,2 - 0.6 (a) Determine the pmf of To-X1 + X2. (b) Calculate Te "T." How does it relate to , the population mean? "T." (c) Calculate e How does it relate to , the population variance? (4) Let X3 and Xg be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the first day. With To- the sum of all four Xs, what now are the values of E(T) and VT)? E(T)- V(T)- (e) Referring back to (d), what are the values of P(To - 8) and P(T, 2 7) (Hint: Don't even think of listing all possible outcomes!] P(T, - 8) - P(T, 2 7)=
There are two traffic lights on a commuter's route to and from work. Let X, be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning from work. Suppose that these two variables are independent, each with the pmf given in the accompanying table (so X, X2 is a random sample of size n- 2). x o 1 2 P(x)0.3 0.4 0.3 -1,2 - 0.6 (a) Determine the pmf of To-X1 + X2. (b) Calculate Te "T." How does it relate to , the population mean? "T." (c) Calculate e How does it relate to , the population variance? (4) Let X3 and Xg be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the first day. With To- the sum of all four Xs, what now are the values of E(T) and VT)? E(T)- V(T)- (e) Referring back to (d), what are the values of P(To - 8) and P(T, 2 7) (Hint: Don't even think of listing all possible outcomes!] P(T, - 8) - P(T, 2 7)=
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 Exercise on Probability and Statistics
### Problem Context:
There are two traffic lights on a commuter’s route to and from work. Let \( X_1 \) be the number of lights at which the commuter must stop on the way to work, and \( X_2 \) be the number of lights at which he must stop when returning from work. These two variables are independent, each with the probability mass function (pmf) given in the table:
| \(X_1\) | 0 | 1 | 2 |
|---------|-----|-----|-----|
| \(p(x_1)\) | 0.3 | 0.4 | 0.3 |
Random sample size \( n = 2 \).
Given:
- \( \mu = 1 \)
- \( \sigma^2 = 0.6 \)
### Questions:
#### (a) Determine the pmf of \( T_0 = X_1 + X_2 \).
| \( t_0 \) | 0 | 1 | 2 | 3 | 4 |
|-----------|---|---|---|---|---|
| \( p(t_0) \) | | | | | |
#### (b) Calculate \( \mu_{T_0} \).
\( \mu_{T_0} = \)
**How does it relate to \( \mu \), the population mean?**
\( \mu_{T_0} = \, \mu \)
#### (c) Calculate \( \sigma_{T_0}^2 \).
\( \sigma_{T_0}^2 = \)
**How does it relate to \( \sigma^2 \), the population variance?**
\( \sigma_{T_0}^2 = \, \sigma^2 \)
#### (d) Let \( X_3 \) and \( X_4 \) be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the first day.
With \( T_0 = \) the sum of all four \( X_i's \), what are the values of \( E(T_0) \) and \( V(T_0) \)?
\[ E(T_0) = \]
\[ V(T_0) = \]
#### (e) Referring back to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94b64eaa-977b-4822-b8ab-e1a6b947c2d1%2Fe1c31990-04b1-4dad-8ab3-5c8a35f61743%2F5oqsrp_processed.png&w=3840&q=75)
Transcribed Image Text:## Educational Exercise on Probability and Statistics
### Problem Context:
There are two traffic lights on a commuter’s route to and from work. Let \( X_1 \) be the number of lights at which the commuter must stop on the way to work, and \( X_2 \) be the number of lights at which he must stop when returning from work. These two variables are independent, each with the probability mass function (pmf) given in the table:
| \(X_1\) | 0 | 1 | 2 |
|---------|-----|-----|-----|
| \(p(x_1)\) | 0.3 | 0.4 | 0.3 |
Random sample size \( n = 2 \).
Given:
- \( \mu = 1 \)
- \( \sigma^2 = 0.6 \)
### Questions:
#### (a) Determine the pmf of \( T_0 = X_1 + X_2 \).
| \( t_0 \) | 0 | 1 | 2 | 3 | 4 |
|-----------|---|---|---|---|---|
| \( p(t_0) \) | | | | | |
#### (b) Calculate \( \mu_{T_0} \).
\( \mu_{T_0} = \)
**How does it relate to \( \mu \), the population mean?**
\( \mu_{T_0} = \, \mu \)
#### (c) Calculate \( \sigma_{T_0}^2 \).
\( \sigma_{T_0}^2 = \)
**How does it relate to \( \sigma^2 \), the population variance?**
\( \sigma_{T_0}^2 = \, \sigma^2 \)
#### (d) Let \( X_3 \) and \( X_4 \) be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the first day.
With \( T_0 = \) the sum of all four \( X_i's \), what are the values of \( E(T_0) \) and \( V(T_0) \)?
\[ E(T_0) = \]
\[ V(T_0) = \]
#### (e) Referring back to
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