The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a continuous rv X with the following pdf. - f(x) = ) = {^ [¹ − (x − 3)²] 2≤x≤4 otherwise 0 (a) Sketch the graph of f(x). f(x) f(x) 1.0 0.8 0.6 0.4 0.2 X X 4.5 2.0 2.5 3.0 3.5 4.0 2.5 2.0 3.5 3.0 4.0 4.5 O O f(x) f(x) 1.0 1.0 0.8 0.8 0.6 0.6 Lh 0.4 0.4 0.2 0.2 X X 2.0 2.5 3.0 3.5 4.0 4.5 2.0 2.5 O 3.0 3.5 4.0 4.5 O (b) Find the value of k. (c) What is the probability that the actual tracking weight is greater than the prescribed weight? 1.0 0.8 0.6 0.4 0.2
The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a continuous rv X with the following pdf. - f(x) = ) = {^ [¹ − (x − 3)²] 2≤x≤4 otherwise 0 (a) Sketch the graph of f(x). f(x) f(x) 1.0 0.8 0.6 0.4 0.2 X X 4.5 2.0 2.5 3.0 3.5 4.0 2.5 2.0 3.5 3.0 4.0 4.5 O O f(x) f(x) 1.0 1.0 0.8 0.8 0.6 0.6 Lh 0.4 0.4 0.2 0.2 X X 2.0 2.5 3.0 3.5 4.0 4.5 2.0 2.5 O 3.0 3.5 4.0 4.5 O (b) Find the value of k. (c) What is the probability that the actual tracking weight is greater than the prescribed weight? 1.0 0.8 0.6 0.4 0.2
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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![### Probability and Statistics: Continuous Random Variables
**Context:** The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a continuous random variable \(X\) with the following probability density function (pdf):
\[
f(x) =
\begin{cases}
k[1 - (x - 3)^2] & \text{for } 2 \le x \le 4 \\
0 & \text{otherwise}
\end{cases}
\]
#### (a) Sketch the graph of \( f(x) \).
There are four sketches provided, but only one is correct. The graph representing \( f(x) \) starts at \( x = 2 \), peaks at \( x = 3 \), and descends back to zero at \( x = 4 \). Each axis is labeled "f(x)" for the vertical and "x" for the horizontal, with the range specified between 2 and 4.
Below is a detailed explanation of these sketches:
- The plot has a parabolic shape between \( x = 2 \) and \( x = 4 \).
- The maximum value is achieved at \( x = 3 \), making it the vertex of the parabola.
- The parabolic curve descends symmetrically on both sides of the vertex.
To determine which sketch to choose, look for the one that peaks at \( x = 3 \) and spans from \( x = 2 \) to \( x = 4 \).
#### (b) Find the value of \( k \).
To ensure that \( f(x) \) is a valid probability density function, it must integrate to 1 over the interval from 2 to 4. This requires setting up and solving the integral:
\[
\int_{2}^{4} k[1 - (x - 3)^2] \, dx = 1
\]
After evaluating the integral, find \( k \).
#### (c) What is the probability that the actual tracking weight is greater than the prescribed weight?
Evaluate the probability \( P(X > 3) \) by integrating the pdf from 3 to 4:
\[
P(X > 3) = \int_{3}^{4} f(x) \, dx
\]
#### (d) What is the probability that the actual weight is within 0.5 g](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda5a8041-aeed-45a6-a7fc-c4165c2d9a1f%2F5b613284-f68e-45d9-8438-74073b9d5c7f%2Fmi95bl5_processed.png&w=3840&q=75)
Transcribed Image Text:### Probability and Statistics: Continuous Random Variables
**Context:** The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a continuous random variable \(X\) with the following probability density function (pdf):
\[
f(x) =
\begin{cases}
k[1 - (x - 3)^2] & \text{for } 2 \le x \le 4 \\
0 & \text{otherwise}
\end{cases}
\]
#### (a) Sketch the graph of \( f(x) \).
There are four sketches provided, but only one is correct. The graph representing \( f(x) \) starts at \( x = 2 \), peaks at \( x = 3 \), and descends back to zero at \( x = 4 \). Each axis is labeled "f(x)" for the vertical and "x" for the horizontal, with the range specified between 2 and 4.
Below is a detailed explanation of these sketches:
- The plot has a parabolic shape between \( x = 2 \) and \( x = 4 \).
- The maximum value is achieved at \( x = 3 \), making it the vertex of the parabola.
- The parabolic curve descends symmetrically on both sides of the vertex.
To determine which sketch to choose, look for the one that peaks at \( x = 3 \) and spans from \( x = 2 \) to \( x = 4 \).
#### (b) Find the value of \( k \).
To ensure that \( f(x) \) is a valid probability density function, it must integrate to 1 over the interval from 2 to 4. This requires setting up and solving the integral:
\[
\int_{2}^{4} k[1 - (x - 3)^2] \, dx = 1
\]
After evaluating the integral, find \( k \).
#### (c) What is the probability that the actual tracking weight is greater than the prescribed weight?
Evaluate the probability \( P(X > 3) \) by integrating the pdf from 3 to 4:
\[
P(X > 3) = \int_{3}^{4} f(x) \, dx
\]
#### (d) What is the probability that the actual weight is within 0.5 g
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