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.
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Chapter1: Starting With Matlab
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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 \( \alpha \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2Feed69cfd-7cd4-4be7-a4f8-7e546be4cc0f%2Fkow5kac_processed.png&w=3840&q=75)
Transcribed Image Text:**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 Function
A 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2Feed69cfd-7cd4-4be7-a4f8-7e546be4cc0f%2Ftftxo49_processed.png&w=3840&q=75)
Transcribed Image Text:### Section 6.17: Power Family Distribution Function
A 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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