Please solve a, b, c, d, and e.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Question
Please solve a, b, c, d, and e.
![(b) Find the value of \( k \).
\[ \boxed{} \]
(c) What is the probability that the actual tracking weight is greater than the prescribed weight?
\[ \boxed{} \]
(d) What is the probability that the actual weight is within \( 0.2 \, g \) of the prescribed weight? (Round your answer to four decimal places.)
\[ \boxed{} \]
(e) What is the probability that the actual weight differs from the prescribed weight by more than \( 0.3 \, g \)? (Round your answer to four decimal places.)
\[ \boxed{} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7afdb6c4-3a61-4c47-91da-dfd2e9ec1a67%2Fed56b961-dbe4-49ef-8e58-f8ba83c15ec6%2Fz16tq_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Find the value of \( k \).
\[ \boxed{} \]
(c) What is the probability that the actual tracking weight is greater than the prescribed weight?
\[ \boxed{} \]
(d) What is the probability that the actual weight is within \( 0.2 \, g \) of the prescribed weight? (Round your answer to four decimal places.)
\[ \boxed{} \]
(e) What is the probability that the actual weight differs from the prescribed weight by more than \( 0.3 \, g \)? (Round your answer to four decimal places.)
\[ \boxed{} \]
![**Educational Content on Probability Density Function of a Stereo Cartridge Tracking Weight**
The actual tracking weight of a stereo cartridge, set to track at 3 grams on a particular changer, can be modeled as a continuous random variable \( X \) with the following probability density function (pdf):
\[
f(x) =
\begin{cases}
k[1 - (x - 3)^2] & \text{for } 2 \leq x \leq 4 \\
0 & \text{otherwise}
\end{cases}
\]
**Graph Sketching**
(a) The pdf \( f(x) \) needs to be sketched. Four different plots are presented, each depicting \( f(x) \) on the interval from \( x = 2.0 \) to \( x = 4.5 \). Each plot displays a curve representing the function within the specified domain.
- The function is a quadratic expression, with the peak at \( x = 3 \) due to the term \( [1 - (x - 3)^2] \), reflecting a downward-opening parabola.
- The value of the pdf is zero outside the interval [2, 4].
**Calculations**
(b) **Finding the value of \( k \):**
To ensure that the total probability is 1, integrate \( f(x) \) over the interval [2, 4] and solve for \( k \).
(c) **Probability of Actual Tracking Weight Greater Than Prescribed Weight:**
Calculate the probability that the actual tracking weight is greater than 3 grams using the calculated function and the appropriate interval.
(d) **Probability Within 0.2 grams of the Prescribed Weight:**
Determine the probability that the actual weight falls within 0.2 grams of the prescribed 3 grams, i.e., the interval [2.8, 3.2]. Round the answer to four decimal places.
Each section builds upon understanding the pdf and applying integration to calculate key probabilities for the stereo cartridge’s tracking weight.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7afdb6c4-3a61-4c47-91da-dfd2e9ec1a67%2Fed56b961-dbe4-49ef-8e58-f8ba83c15ec6%2Figjoh7c_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Content on Probability Density Function of a Stereo Cartridge Tracking Weight**
The actual tracking weight of a stereo cartridge, set to track at 3 grams on a particular changer, can be modeled as a continuous random variable \( X \) with the following probability density function (pdf):
\[
f(x) =
\begin{cases}
k[1 - (x - 3)^2] & \text{for } 2 \leq x \leq 4 \\
0 & \text{otherwise}
\end{cases}
\]
**Graph Sketching**
(a) The pdf \( f(x) \) needs to be sketched. Four different plots are presented, each depicting \( f(x) \) on the interval from \( x = 2.0 \) to \( x = 4.5 \). Each plot displays a curve representing the function within the specified domain.
- The function is a quadratic expression, with the peak at \( x = 3 \) due to the term \( [1 - (x - 3)^2] \), reflecting a downward-opening parabola.
- The value of the pdf is zero outside the interval [2, 4].
**Calculations**
(b) **Finding the value of \( k \):**
To ensure that the total probability is 1, integrate \( f(x) \) over the interval [2, 4] and solve for \( k \).
(c) **Probability of Actual Tracking Weight Greater Than Prescribed Weight:**
Calculate the probability that the actual tracking weight is greater than 3 grams using the calculated function and the appropriate interval.
(d) **Probability Within 0.2 grams of the Prescribed Weight:**
Determine the probability that the actual weight falls within 0.2 grams of the prescribed 3 grams, i.e., the interval [2.8, 3.2]. Round the answer to four decimal places.
Each section builds upon understanding the pdf and applying integration to calculate key probabilities for the stereo cartridge’s tracking weight.
Expert Solution
Step 1
Hi! Thank you for the question, As per the honor code, we are allowed to answer three sub-parts at a time so we are answering the first three as you have not mentioned which of these you are looking for. Please re-submit the question separately for the remaining sub-parts.
The with pdf,
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON