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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)
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- a. b. Let X have a gamma distribution with a=2 and 3-2. What is P(X>2)? Let X have a log-normal distribution with u-3.5 and o=1.2. Find P(50Let T represent the lifetime in years of a part which follows a Weibull distribution with shape 2 and scale 5. (a) What is E(T )? Make sure to simplify the gamma function in terms of pi.(b) What is V(T )? Make sure to simplify the gamma function in terms of pi.Show clearly all working with detailed explanations*2. Let log Y the median of Y). ~ N (u, o2). Find the distribution of Y and show that P(Y < eH) = 0.5 (i.e., el is11 Assume the GPAS of high school students, X, follows a continuous uniform on the domain 1Using a suitable linearization to approximate 1/1.02 show that (i)The error is at most 0.0004. That is E (0.0096, 1.0004)Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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