Let Y have probability density function a Show that Y has

Chapter 8, Problem 44E

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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

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

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