/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q1E Question: In a clinical trial, l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q1E (page 416)

Question: In a clinical trial, let the probability of successful outcome 胃 have a prior distribution that is the uniform distribution on the interval\(\left[ {0,1} \right]\), which is also the beta distribution with parameters 1 and 1. Suppose that the first patient has a successful outcome. Find the Bayes estimates of 胃 that would be obtained for both the squared error and absolute error loss functions.

Short Answer

Expert verified

The mean of the posterior distribution is 2/3, which would be the Bayes estimate under squared error loss and Median of the posterior distribution is 0.7071.

Step by step solution

01

Given information

It is given that the probability of successful outcome \(\theta \)has a uniform distribution on the interval\(\left( {0,1} \right)\) which is also the beta distribution with parameter 1 and 1.

02

Calculating Bayes estimates of \(\theta \) 

The posterior distribution of\(\theta \)would be the beta distribution with parameter 1 and 1. The mean of the posterior distribution is 2/3, which would be the Bayes estimate under squared error loss.

The median of the posterior distribution would be the Bayes estimate under absolute error loss.

To find the median, write the c.d.f as

\(\begin{array}{c}F\left( \theta \right) = \int\limits_0^\theta {2t\;dt\;} \\ = {\theta ^2}\end{array}\)

For\(0 < \theta < 1\)

The quantile function is then\({F^{ - 1}}\left( p \right) = {p^{1/2}}\)

So the median is, \({\left( {\frac{1}{2}} \right)^{1/2}} = 0.7071\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Assume that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from a distribution that belongs to an exponential family of distributions as defined in Exercise 23 of Sec. 7.3. Prove that\({\bf{T = }}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{\bf{d}}\left( {{{\bf{X}}_{\bf{i}}}} \right)} \) is a sufficient statistic for 胃.

Suppose that a single observation X is to be taken from the uniform distribution on the interval \(\left[ {{\bf{\theta - }}\frac{{\bf{1}}}{{\bf{2}}}{\bf{,\theta + }}\frac{{\bf{1}}}{{\bf{2}}}} \right]\), the value of 胃 is unknown, and the prior distribution of 胃 is the uniform distribution on the interval [10, 20]. If the observed value of X is 12, what is the posterior distribution of 胃?

Show that the family of discrete uniform distributions on the sets of integers \(\left\{ {0,1...\theta } \right\}\) for \(\theta \) a nonnegative integer is not an exponential family as defined in Exercise

Question: Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from the uniform distribution on the interval \(\left( {{{\bf{\theta }}_{\bf{1}}}{\bf{,}}{{\bf{\theta }}_{\bf{2}}}} \right)\) , where both \({{\bf{\theta }}_{\bf{1}}}\)and \({{\bf{\theta }}_{\bf{2}}}\) are unknown . Find the M.L.E.鈥檚 \({{\bf{\theta }}_{\bf{1}}}\) and \({{\bf{\theta }}_{\bf{2}}}\).

Consider again the problem described in Exercise 6, and assume the same prior distribution of 胃. Suppose now, however, that instead of selecting a random sample of eight items from the lot, we perform the following experiment: Items from the lot are selected at random one by one until exactly three defectives have been found. If we find that we must select a total of eight items in this experiment, what is the posterior distribution of 胃 at the end of the experiment?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.