Question

In: Statistics and Probability

3. (Not Bayesian) Let Y have pdf fY (y, θ) = 2(θ − y)/θ2 if 0...

3. (Not Bayesian) Let Y have pdf fY (y, θ) =
2(θ − y)/θ2
if 0 < y < θ
0 otherwise . You are
going to construct a confidence interval for θ based on a single observation Y .
(a) Show that Y/θ is pivotal quantity.
(b) Suppose Y = 6. Find (numerically) a 10% confidence interval for θ .

Expert Solution

The given pdf of Y is

Let . When y=0, u=0 and when

Then and and

the pdf of U is given by

So the pdf of does not depend on and hence is a pivotal quantity.

Let a be the point such that

Then

implies

Therefore will be a 10% confidence interval for .

Now we will find the value of a.

We know P(a<U<1)=0.1

That is

or

Solving this quadratic equation we get the value of a as

So and the 10% confidence interval becomes

orchestra answered 2 years ago

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