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3r² 1. The radius R of a certain disk is a continuous random variable with pdf fr(r) = 125' (a) What is the probability that 3 < R < 4? (b) Find and simplify a formula for the cdf Fr(r) of the random variable R. (c) What is the expected value E(R)? (0,0) where 0

3r² 1. The radius R of a certain disk is a continuous random variable with pdf fr(r) = 125' (a) What is the probability that 3 < R < 4? (b) Find and simplify a formula for the cdf Fr(r) of the random variable R. (c) What is the expected value E(R)? (0,0) where 0

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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The image presents a problem about a continuous random variable associated with the radius \( R \) of a disk. Here's the transcription and analysis of its content: 1. The radius \( R \) of a certain disk is a continuous random variable with probability density function (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 cumulative distribution function (cdf) \( F_R(r) \) of the random variable \( R \). (c) What is the expected value \( E(R) \)? There is also a diagram illustrating a circle with center at \((0,0)\) and radius \( R \). The point \((R,0)\) lies on the circumference, indicating the radius distance from the center to a point on the edge of the circle. This image explores the mathematical concepts of probability, probability density functions, cumulative distribution functions, and expected values in the context of probability theory for continuous random variables.
Transcribed Image Text:The image presents a problem about a continuous random variable associated with the radius \( R \) of a disk. Here's the transcription and analysis of its content: 1. The radius \( R \) of a certain disk is a continuous random variable with probability density function (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 cumulative distribution function (cdf) \( F_R(r) \) of the random variable \( R \). (c) What is the expected value \( E(R) \)? There is also a diagram illustrating a circle with center at \((0,0)\) and radius \( R \). The point \((R,0)\) lies on the circumference, indicating the radius distance from the center to a point on the edge of the circle. This image explores the mathematical concepts of probability, probability density functions, cumulative distribution functions, and expected values in the context of probability theory for continuous random variables.
(d) What is the expected area of the disk of radius R? *(Think about this carefully!)* --- **Explanation of the Concept:** - A disk, in geometric terms, is a shape defined in a two-dimensional space with all points at a distance less than or equal to a given radius, R, from a fixed point (the center). - The formula for the area of a circle (which is equivalent to a disk) is given by \(A = \pi R^2\), where \(A\) is the area and \(R\) is the radius. **Considerations:** When approaching problems involving area and geometry, it's important to consider: - The units involved, ensuring consistency (e.g., all measurements should be in the same unit system). - The value of \(\pi\), which is approximately 3.14159, can have implications for precision in calculations, especially in educational or applied contexts. **Educational Insight:** This problem encourages conceptual thinking about geometric properties and their applications. Understanding the relationship between radius, area, and the constant \(\pi\) is crucial for solving a variety of mathematical and real-world problems.
Transcribed Image Text:(d) What is the expected area of the disk of radius R? *(Think about this carefully!)* --- **Explanation of the Concept:** - A disk, in geometric terms, is a shape defined in a two-dimensional space with all points at a distance less than or equal to a given radius, R, from a fixed point (the center). - The formula for the area of a circle (which is equivalent to a disk) is given by \(A = \pi R^2\), where \(A\) is the area and \(R\) is the radius. **Considerations:** When approaching problems involving area and geometry, it's important to consider: - The units involved, ensuring consistency (e.g., all measurements should be in the same unit system). - The value of \(\pi\), which is approximately 3.14159, can have implications for precision in calculations, especially in educational or applied contexts. **Educational Insight:** This problem encourages conceptual thinking about geometric properties and their applications. Understanding the relationship between radius, area, and the constant \(\pi\) is crucial for solving a variety of mathematical and real-world problems.
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