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S.1-10. The lifetime (in years) of a manufactured product sY= 5X7, where X has an exponential distribution ih mean 1. Find the cdf and pdf of Y. %3D .1-1

S.1-10. The lifetime (in years) of a manufactured product sY= 5X7, where X has an exponential distribution ih mean 1. Find the cdf and pdf of Y. %3D .1-1

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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5.1-10

**Transcription for Educational Website:** Page Section: --- ### Exercises #### 5.1-9. **Problem:** A sum of $50,000 is invested at a rate \( R \), selected from a uniform distribution on the interval (0.03, 0.07). Once \( R \) is selected, the sum is compounded instantaneously for a year, so that \( X = 50000e^R \) dollars is the amount at the end of that year. **Tasks:** (a) Find the cumulative distribution function (cdf) and probability density function (pdf) of \( X \). (b) Verify that \( X = 50000e^R \) is defined correctly if the compounding is done instantaneously. **Hint:** Divide the year into \( n \) equal parts, calculate the value of the amount at the end of each part, and then take the limit as \( n \rightarrow \infty \). --- #### 5.1-10. **Problem:** The lifetime (in years) of a manufactured product is \( Y = 5X^{0.7} \), where \( X \) has an exponential distribution with mean 1. **Task:** Find the cdf and pdf of \( Y \). --- #### 5.1-11. Statisticians frequently use the **extreme value distribution** given by the cdf: \[ F(x) = 1 - \exp\left[-e^{(x - \theta_1)/\theta_2}\right], \quad -\infty < x < \infty. \] A simple case is when \( \theta_1 = 0 \) and \( \theta_2 = 1 \), giving \[ F(x) = 1 - \exp\left[-e^x\right], \quad -\infty < x < \infty. \] --- **Note:** There are no graphs or diagrams accompanying these exercises.
Transcribed Image Text:**Transcription for Educational Website:** Page Section: --- ### Exercises #### 5.1-9. **Problem:** A sum of $50,000 is invested at a rate \( R \), selected from a uniform distribution on the interval (0.03, 0.07). Once \( R \) is selected, the sum is compounded instantaneously for a year, so that \( X = 50000e^R \) dollars is the amount at the end of that year. **Tasks:** (a) Find the cumulative distribution function (cdf) and probability density function (pdf) of \( X \). (b) Verify that \( X = 50000e^R \) is defined correctly if the compounding is done instantaneously. **Hint:** Divide the year into \( n \) equal parts, calculate the value of the amount at the end of each part, and then take the limit as \( n \rightarrow \infty \). --- #### 5.1-10. **Problem:** The lifetime (in years) of a manufactured product is \( Y = 5X^{0.7} \), where \( X \) has an exponential distribution with mean 1. **Task:** Find the cdf and pdf of \( Y \). --- #### 5.1-11. Statisticians frequently use the **extreme value distribution** given by the cdf: \[ F(x) = 1 - \exp\left[-e^{(x - \theta_1)/\theta_2}\right], \quad -\infty < x < \infty. \] A simple case is when \( \theta_1 = 0 \) and \( \theta_2 = 1 \), giving \[ F(x) = 1 - \exp\left[-e^x\right], \quad -\infty < x < \infty. \] --- **Note:** There are no graphs or diagrams accompanying these exercises.
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