iid *1. Let Y₁,..., Yn uniform(0, 1). Find the distribution of their geometric mean: U = (ÎIx.)* ΠΥ i=1 (Hint: consider first finding the distribution of -n log U.) *2. Let log Y ~ N (μ, o²). Find the distribution of Y and show that P(Y ≤ e¹¹) = 0.5 (i.e., et is the median of Y).
iid *1. Let Y₁,..., Yn uniform(0, 1). Find the distribution of their geometric mean: U = (ÎIx.)* ΠΥ i=1 (Hint: consider first finding the distribution of -n log U.) *2. Let log Y ~ N (μ, o²). Find the distribution of Y and show that P(Y ≤ e¹¹) = 0.5 (i.e., et is the median of Y).
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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**Mathematical Problem Set**
**Problem 1**:
Let \( Y_1, \ldots, Y_n \) be independent and identically distributed (i.i.d) random variables following a uniform distribution over the interval (0, 1). Find the distribution of their geometric mean:
\[ U = \left( \prod_{i=1}^{n} Y_i \right)^{\frac{1}{n}} \]
*(Hint: Consider first finding the distribution of \(-n \log U\).)*
**Problem 2**:
Let \( \log Y \sim \mathcal{N}(\mu, \sigma^2) \). Find the distribution of \( Y \) and show that \( P(Y \leq e^\mu) = 0.5 \) (i.e., \( e^\mu \) is the median of \( Y \)).
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**Detailed Explanation**:
**Problem 1** Explanation:
- \( Y_1, \ldots, Y_n \) are each uniformly distributed random variables between 0 and 1.
- We need to determine the distribution of the geometric mean of these variables.
- The hint suggests that a useful strategy involves understanding the distribution of \(-n \log U\), where \( U \) is the geometric mean.
**Problem 2** Explanation:
- Here, \( \log Y \) is normally distributed with mean \( \mu \) and variance \( \sigma^2 \).
- The task is to find the distribution of \( Y \), which involves an exponential transformation of a normally distributed variable.
- Additionally, one needs to show that the probability \( P(Y \leq e^\mu) = 0.5 \), establishing that \( e^\mu \) is the median of \( Y \).
---
This problem set aims to enhance understanding of the distribution of transformed variables and the geometric mean, as well as the handling of exponential transformations in probability distributions. The hints and preliminary steps are crucial for solving the problems effectively.
---
For any questions or further discussion on these problems, please refer to the instructor or provided mathematical resources.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8952a85a-85f1-4c32-8b45-c088a1def42e%2F6e1697ca-b112-4bd0-8c94-3f36257cb7c4%2F82f0aon_processed.png&w=3840&q=75)
Transcribed Image Text:---
**Mathematical Problem Set**
**Problem 1**:
Let \( Y_1, \ldots, Y_n \) be independent and identically distributed (i.i.d) random variables following a uniform distribution over the interval (0, 1). Find the distribution of their geometric mean:
\[ U = \left( \prod_{i=1}^{n} Y_i \right)^{\frac{1}{n}} \]
*(Hint: Consider first finding the distribution of \(-n \log U\).)*
**Problem 2**:
Let \( \log Y \sim \mathcal{N}(\mu, \sigma^2) \). Find the distribution of \( Y \) and show that \( P(Y \leq e^\mu) = 0.5 \) (i.e., \( e^\mu \) is the median of \( Y \)).
---
**Detailed Explanation**:
**Problem 1** Explanation:
- \( Y_1, \ldots, Y_n \) are each uniformly distributed random variables between 0 and 1.
- We need to determine the distribution of the geometric mean of these variables.
- The hint suggests that a useful strategy involves understanding the distribution of \(-n \log U\), where \( U \) is the geometric mean.
**Problem 2** Explanation:
- Here, \( \log Y \) is normally distributed with mean \( \mu \) and variance \( \sigma^2 \).
- The task is to find the distribution of \( Y \), which involves an exponential transformation of a normally distributed variable.
- Additionally, one needs to show that the probability \( P(Y \leq e^\mu) = 0.5 \), establishing that \( e^\mu \) is the median of \( Y \).
---
This problem set aims to enhance understanding of the distribution of transformed variables and the geometric mean, as well as the handling of exponential transformations in probability distributions. The hints and preliminary steps are crucial for solving the problems effectively.
---
For any questions or further discussion on these problems, please refer to the instructor or provided mathematical resources.
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