Answer 9.43 **Exercise 9.43**Consider the random variables \( Y_1, Y_2, \ldots, Y_n \) which are independent and identically distributed from a power family distribution with parameters \( \alpha \) and \( \theta \). Given the result from Exercise 6.17, suppose \( \alpha, \theta > 0 \).The probability density function is given by:\[f(y \mid \alpha, \theta) = \begin{cases} \frac{\alpha y^{\alpha-1}}{\theta^{\alpha}}, & 0 \leq y \leq \theta, \\0, & \text{elsewhere}.\end{cases}\]If \( \theta \) is known, demonstrate that \(\prod_{i=1}^{n} Y_i\) is sufficient for \( \alpha \). ### Section 6.17: Power Family Distribution FunctionA member of the power family of distributions is defined by its distribution function:\[F(y) = \begin{cases} 0, & y < 0, \\\left( \frac{y}{\theta} \right)^\alpha, & 0 \leq y \leq \theta, \\1, & y > \theta \end{cases}\]Parameters:- \( \alpha, \theta > 0 \)**Task Instructions:**a. **Find the Density Function:** - Derive the probability density function (pdf) from the given distribution function \( F(y) \).b. **Transformation for Uniform Distribution:** - For fixed values of \( \alpha \) and \( \theta \), find a transformation \( G(U) \) such that \( G(U) \) has a distribution function \( F \) when \( U \) possesses a uniform (0, 1) distribution.c. **Random Sample Transformation:** - Consider a random sample of size 5 from a uniform distribution within the interval \([0, 1]\) resulting in the values: 0.2700, 0.6901, 0.1413, 0.1523, 0.3609. - Use the transformation derived in part (b) to calculate values associated with a random variable using the power family distribution with \( \alpha = 2 \) and \( \theta = 4 \).**Note:** This section provides a framework for understanding the behavior and applications of the power family of distributions through transformations and practical sampling.

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9.43 Let Y₁, Y₂,..., Y₁ denote independent and identically distributed random variables from a power family distribution with parameters a and 0. Then, by the result in Exercise 6.17, if a, 0 > 0, f(y|a, 0) = [aya-1/0°, 0₁. If is known, show that I Y; is sufficient for a. 0≤ y ≤0, elsewhere.

9.43 Let Y₁, Y₂,..., Y₁ denote independent and identically distributed random variables from a power family distribution with parameters a and 0. Then, by the result in Exercise 6.17, if a, 0 > 0, f(y|a, 0) = [aya-1/0°, 0₁. If is known, show that I Y; is sufficient for a. 0≤ y ≤0, elsewhere.

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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