*1. Let Y1,...,Yn * uniform(0, 1). Find the distribution of their geometric mean: U = (11: II (Hint: consider first finding the distribution of -n log U.)
*1. Let Y1,...,Yn * uniform(0, 1). Find the distribution of their geometric mean: U = (11: II (Hint: consider first finding the distribution of -n log U.)
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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![1. Let \( Y_1, \ldots, Y_n \stackrel{\text{iid}}{\sim} \text{uniform}(0, 1) \). 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 \(-n \log U\).)
2. Let \(\log Y \sim N (\mu, \sigma^2) \). Find the distribution of \( Y \) and show that \( P(Y \leq e^\mu) = 0.5 \) (i.e., \( e^\mu \) is the median of \( Y \)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F544a9f38-3e68-4113-87a7-993914a696cb%2Fa7d08730-11f4-45e1-8789-62c30430fef8%2Fjna6qmh_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let \( Y_1, \ldots, Y_n \stackrel{\text{iid}}{\sim} \text{uniform}(0, 1) \). 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 \(-n \log U\).)
2. Let \(\log Y \sim N (\mu, \sigma^2) \). 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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