which the commuter must stop on his way to work, and X, be the number of lights at which he must stop when returning from work. Suppose that these There are two traffic lights on a commuter's route to and from work. Let X, be the number of lights are independent, each with the pmf given in the accompanying table (so X,, X, is a random sample of size n = 2). 1 2 P(x,) 0.4 0.2 0.4 H = 1, o = 0.8 (a) Determine the pmf of T, = X, + X2- 1 3 4.
which the commuter must stop on his way to work, and X, be the number of lights at which he must stop when returning from work. Suppose that these There are two traffic lights on a commuter's route to and from work. Let X, be the number of lights are independent, each with the pmf given in the accompanying table (so X,, X, is a random sample of size n = 2). 1 2 P(x,) 0.4 0.2 0.4 H = 1, o = 0.8 (a) Determine the pmf of T, = X, + X2- 1 3 4.
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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Please show and explain the steps! Solve only (d) and (e)!
![The image presents a statistical problem involving a commuter and traffic lights. There are two traffic lights on the commuter's route to and from work. We define \( X_1 \) as the number of lights at which the commuter must stop on the way to work, and \( X_2 \) for the return trip. Both variables are independent, with a probability mass function (pmf) provided as follows:
\[
\begin{array}{c|ccc}
x_1 & 0 & 1 & 2 \\
\hline
p(x_1) & 0.4 & 0.2 & 0.4 \\
\end{array}
\]
The mean \( \mu = 1 \) and variance \( \sigma^2 = 0.8 \).
**Questions and Tasks:**
(a) **Determine the pmf of \( T_0 = X_1 + X_2 \):**
The values for \( t_0 \) range from 0 to 4, and boxes are provided to fill in corresponding probabilities \( p(t_0) \).
(b) **Calculate \( \mu_{T_0} \):**
This value should relate to \( \mu \), the population mean.
- \( \mu_{T_0} = \_\_\_\_ \)
- \( \mu_{T_0} = \_\_\_\_ \cdot \mu \)
(c) **Calculate \( \sigma^2_{T_0} \):**
This value should relate to \( \sigma^2 \), the population variance.
- \( \sigma^2_{T_0} = \_\_\_\_ \)
- \( \sigma^2_{T_0} = \_\_\_\_ \cdot \sigma^2 \)
(d) **Consider a second day with variables \( X_3 \) and \( X_4 \):**
On the second day, assume independence of the first day. Let \( T_o \) be the sum of all four \( X \)'s; calculate:
- \( E(T_o) = \_\_\_\_ \)
- \( V(T_o) = \_\_\_\_ \)
(e) **Determine probabilities related to \( P(T_o = 8) \) and \( P(T_o \geq 7) \):**
The problem hints not to list all possible outcomes when calculating these](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc32f563a-6b55-4123-a4c3-1254c39a7403%2F9953e564-4f4d-40ba-bb49-5c7104a9220a%2Fu8hynhj_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents a statistical problem involving a commuter and traffic lights. There are two traffic lights on the commuter's route to and from work. We define \( X_1 \) as the number of lights at which the commuter must stop on the way to work, and \( X_2 \) for the return trip. Both variables are independent, with a probability mass function (pmf) provided as follows:
\[
\begin{array}{c|ccc}
x_1 & 0 & 1 & 2 \\
\hline
p(x_1) & 0.4 & 0.2 & 0.4 \\
\end{array}
\]
The mean \( \mu = 1 \) and variance \( \sigma^2 = 0.8 \).
**Questions and Tasks:**
(a) **Determine the pmf of \( T_0 = X_1 + X_2 \):**
The values for \( t_0 \) range from 0 to 4, and boxes are provided to fill in corresponding probabilities \( p(t_0) \).
(b) **Calculate \( \mu_{T_0} \):**
This value should relate to \( \mu \), the population mean.
- \( \mu_{T_0} = \_\_\_\_ \)
- \( \mu_{T_0} = \_\_\_\_ \cdot \mu \)
(c) **Calculate \( \sigma^2_{T_0} \):**
This value should relate to \( \sigma^2 \), the population variance.
- \( \sigma^2_{T_0} = \_\_\_\_ \)
- \( \sigma^2_{T_0} = \_\_\_\_ \cdot \sigma^2 \)
(d) **Consider a second day with variables \( X_3 \) and \( X_4 \):**
On the second day, assume independence of the first day. Let \( T_o \) be the sum of all four \( X \)'s; calculate:
- \( E(T_o) = \_\_\_\_ \)
- \( V(T_o) = \_\_\_\_ \)
(e) **Determine probabilities related to \( P(T_o = 8) \) and \( P(T_o \geq 7) \):**
The problem hints not to list all possible outcomes when calculating these
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