Solution Found!
Let Y have probability density function a Show that Y has
Chapter 8, Problem 44E(choose chapter or problem)
Let Y have probability density function
\( \ f_{Y}(y)=\left\{\begin{array}{ll} \frac{2(\theta-y)}{\theta^{2}}, & 0<y<\theta \\ 0, & \text
{ elsewhere } \end{array}\right. \)
a. Show that Y has distribution function
\(F_{Y}(y)=\left\{\begin{array}{ll} 0, & y \leq 0, \\ \frac{2 y}{\theta}-\frac{y^{2}}{\theta^{2}}, & 0<y<\theta, \\ 1, & y \geq \theta . \end{array}\right.\)
b. Show that \(\mathrm{Y} / \theta\) is a pivotal quantity.
c. Use the pivotal quantity from part (b) to find a 90% lower confidence limit for \(\theta\)
Equation Transcription:
{
{
Text Transcription:.
\(f_{Y}(y)={2(\theta-y)\theta^2, & 0<y<\theta \ 0, & elsewhere
\(F_{Y}(y)= 0, & y \leq 0, \ \frac{2 y\theta-\frac{y^2\theta^2, & 0<y<\theta, \\ 1, & y \geq \theta .
Y / \theta
\theta
Questions & Answers
QUESTION:
Let Y have probability density function
\( \ f_{Y}(y)=\left\{\begin{array}{ll} \frac{2(\theta-y)}{\theta^{2}}, & 0<y<\theta \\ 0, & \text
{ elsewhere } \end{array}\right. \)
a. Show that Y has distribution function
\(F_{Y}(y)=\left\{\begin{array}{ll} 0, & y \leq 0, \\ \frac{2 y}{\theta}-\frac{y^{2}}{\theta^{2}}, & 0<y<\theta, \\ 1, & y \geq \theta . \end{array}\right.\)
b. Show that \(\mathrm{Y} / \theta\) is a pivotal quantity.
c. Use the pivotal quantity from part (b) to find a 90% lower confidence limit for \(\theta\)
Equation Transcription:
{
{
Text Transcription:.
\(f_{Y}(y)={2(\theta-y)\theta^2, & 0<y<\theta \ 0, & elsewhere
\(F_{Y}(y)= 0, & y \leq 0, \ \frac{2 y\theta-\frac{y^2\theta^2, & 0<y<\theta, \\ 1, & y \geq \theta .
Y / \theta
\theta
ANSWER:Step 1 of 5
Given data
Let has probability density function
(a)
has probability distribution function
Now
Case 1
If then as