---**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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Answered: iid *1. Let Y₁,..., Yn uniform(0, 1).…

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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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).