Chapter 11, Bayesian Statistics Video Solutions, Introduction to Mathematical Statistics

Problem 1

Let $Y$ have a binomial distribution in which $n=20$ and $p=\theta$. The prior probabilities on $\theta$ are $P(\theta=0.3)=2 / 3$ and $P(\theta=0.5)=1 / 3$. If $y=9$, what are the posterior probabilities for $\theta=0.3$ and $\theta=0.5$ ?

Amany Waheeb

Numerade Educator

02:03

Problem 2

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a distribution that is $b(1, \theta)$. Let the prior of $\Theta$ be a beta one with parameters $\alpha$ and $\beta .$ Show that the posterior $\operatorname{pdf} k\left(\theta \mid x_{1}, x_{2}, \ldots, x_{n}\right)$ is exactly the same as $k(\theta \mid y)$ given in Example 11.1.2.

Manik Pulyani

Numerade Educator

03:52

Problem 3

Let $X_{1}, X_{2}, \ldots, X_{n}$ denote a random sample from a distribution that is $N\left(\theta, \sigma^{2}\right)$, where $-\infty<\theta<\infty$ and $\sigma^{2}$ is a given positive number. Let $Y=\bar{X}$ denote the mean of the random sample. Take the loss function to be $\mathcal{L}[\theta, \delta(y)]=|\theta-\delta(y)|$. If $\theta$ is an observed value of the random variable $\Theta$ that is $N\left(\mu, \tau^{2}\right)$, where $\tau^{2}>0$ and $\mu$ are known numbers, find the Bayes solution $\delta(y)$ for a point estimate $\theta$.

Khoobchandra Agrawal

Numerade Educator

01:20

Problem 4

Let $X_{1}, X_{2}, \ldots, X_{n}$ denote a random sample from a Poisson distribution with mean $\theta, 0<\theta<\infty$. Let $Y=\sum_{1}^{n} X_{i} .$ Use the loss function $\mathcal{L}[\theta, \delta(y)]=$ $[\theta-\delta(y)]^{2}$. Let $\theta$ be an observed value of the random variable $\Theta$. If $\Theta$ has the prior $\operatorname{pdf} h(\theta)=\theta^{\alpha-1} e^{-\theta / \beta} / \Gamma(\alpha) \beta^{\alpha}$, for $0<\theta<\infty$, zero elsewhere, where $\alpha>0, \beta>0$
are known numbers, find the Bayes solution $\delta(y)$ for a point estimate for $\theta$.

Manik Pulyani

Numerade Educator

01:58

Problem 5

Let $Y_{n}$ be the $n$ th order statistic of a random sample of size $n$ from a distribution with pdf $f(x \mid \theta)=1 / \theta, 0<x<\theta$, zero elsewhere. Take the loss function to be $\mathcal{L}[\theta, \delta(y)]=\left[\theta-\delta\left(y_{n}\right)\right]^{2}$. Let $\theta$ be an observed value of the random variable $\Theta$, which has the prior pdf $h(\theta)=\beta \alpha^{\beta} / \theta^{\beta+1}, \alpha<\theta<\infty$, zero elsewhere, with $\alpha>0, \beta>0$. Find the Bayes solution $\delta\left(y_{n}\right)$ for a point estimate of $\theta$.

Manik Pulyani

Numerade Educator

05:10

Problem 6

Let $Y_{1}$ and $Y_{2}$ be statistics that have a trinomial distribution with parameters $n, \theta_{1}$, and $\theta_{2}$. Here $\theta_{1}$ and $\theta_{2}$ are observed values of the random variables $\Theta_{1}$ and $\Theta_{2}$, which have a Dirichlet distribution with known parameters $\alpha_{1}, \alpha_{2}$, and $\alpha_{3}$; see expression (3.3.10). Show that the conditional distribution of $\Theta_{1}$ and $\Theta_{2}$ is Dirichlet and determine the conditional means $E\left(\Theta_{1} \mid y_{1}, y_{2}\right)$ and $E\left(\Theta_{2} \mid y_{1}, y_{2}\right)$.

Jacquelinne S. Mejia Sandoval

Numerade Educator

05:01

Problem 7

For Example 11.1.6, obtain the $95 \%$ credible interval for $\theta$. Next obtain the value of the mle for $\theta$ and the $95 \%$ confidence interval for $\theta$ discussed in Chapter 6 .

Raymond Matshanda

Numerade Educator

01:49

Problem 8

In Example 11.1.2, let $n=30, \alpha=10$, and $\beta=5$, so that $\delta(y)=(10+y) / 45$ is the Bayes estimate of $\theta$.
(a) If $Y$ has a binomial distribution $b(30, \theta)$, compute the risk $E\left\{[\theta-\delta(Y)]^{2}\right\}$.
(b) Find values of $\theta$ for which the risk of part (a) is less than $\theta(1-\theta) / 30$, the risk associated with the maximum likelihood estimator $Y / n$ of $\theta$.

Ameer Said

Numerade Educator

01:05

Problem 9

Let $Y_{4}$ be the largest order statistic of a sample of size $n=4$ from a distribution with uniform pdf $f(x ; \theta)=1 / \theta, 0<x<\theta$, zero elsewhere. If the prior pdf of the parameter $g(\theta)=2 / \theta^{3}, 1<\theta<\infty$, zero elsewhere, find the Bayesian estimator $\delta\left(Y_{4}\right)$ of $\theta$, based upon the sufficient statistic $Y_{4}$, using the loss function $\left|\delta\left(y_{4}\right)-\theta\right| .$

Manik Pulyani

Numerade Educator

01:40

Problem 10

Refer to Example 11.2.3; suppose we select $\sigma_{0}^{2}=d \sigma^{2}$, where $\sigma^{2}$ was known in that example. What value do we assign to $d$ so that the variance of posterior is two-thirds the variance of $Y=\bar{X}$, namely, $\sigma^{2} / n ?$

Hunza Gilgit

Numerade Educator