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3r² continuous random variable with pdf fr(r) = where 0 < R < 5. 125' 1. The radius R of a certain disk is a (a) What is the probability that 3 < R < 4?

3r² continuous random variable with pdf fr(r) = where 0 < R < 5. 125' 1. The radius R of a certain disk is a (a) What is the probability that 3 < R < 4?

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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**1. The radius \( R \) of a certain disk is a continuous random variable with pdf \( f_R(r) = \frac{3r^2}{125} \), where \( 0 < R < 5 \).** **(a)** What is the probability that \( 3 < R < 4 \)? **(b)** Find and simplify a formula for the cdf \( F_R(r) \) of the random variable \( R \). **(c)** What is the expected value \( E(R) \)? **Diagram Explanation:** The diagram illustrates a circle with a radius \( R \). The circle is centered at the origin, \((0,0)\), and the point \((R,0)\) marks the endpoint of the radius \( R \) along the x-axis. This visual aids in understanding that \( R \) represents the radius of a disk with given properties.
Transcribed Image Text:**1. The radius \( R \) of a certain disk is a continuous random variable with pdf \( f_R(r) = \frac{3r^2}{125} \), where \( 0 < R < 5 \).** **(a)** What is the probability that \( 3 < R < 4 \)? **(b)** Find and simplify a formula for the cdf \( F_R(r) \) of the random variable \( R \). **(c)** What is the expected value \( E(R) \)? **Diagram Explanation:** The diagram illustrates a circle with a radius \( R \). The circle is centered at the origin, \((0,0)\), and the point \((R,0)\) marks the endpoint of the radius \( R \) along the x-axis. This visual aids in understanding that \( R \) represents the radius of a disk with given properties.
**Question (d):** What is the expected *area* of the disk of radius \( R \)? *(Think about this carefully!)* --- The question above prompts students to consider the mathematical expression for calculating the area of a disk. It encourages careful thinking, possibly hinting at considerations such as integration over probability distributions if the context is probabilistic or geometric aspects in an area calculation. For a simple calculation in Euclidean geometry, the area \( A \) of a disk (or circle) with radius \( R \) is given by the formula: \[ A = \pi R^2 \] If the context involves probability or expectations, students might need to integrate or average over some distribution.
Transcribed Image Text:**Question (d):** What is the expected *area* of the disk of radius \( R \)? *(Think about this carefully!)* --- The question above prompts students to consider the mathematical expression for calculating the area of a disk. It encourages careful thinking, possibly hinting at considerations such as integration over probability distributions if the context is probabilistic or geometric aspects in an area calculation. For a simple calculation in Euclidean geometry, the area \( A \) of a disk (or circle) with radius \( R \) is given by the formula: \[ A = \pi R^2 \] If the context involves probability or expectations, students might need to integrate or average over some distribution.
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