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Please solve a, b, c, d, and e.

Please solve a, b, c, d, and e.

A First Course in Probability (10th Edition)
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...
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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.) \[ \boxed{} \]
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.
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.
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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 continuous random variable X with pdf,

fx=k1-x-32    2≤x≤40                         otherwise

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