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Chapter 11: Problem 1

Suppose \(Y\) has a \(\Gamma(1,1)\) distribution while \(X\) given \(Y\) has the conditional pdf $$ f(x \mid y)=\left\\{\begin{array}{ll} e^{-(x-y)} & 0

Short Answer

Expert verified
To conduct the task, set up the simulations based on given PDFs, estimate \(E(X)\) through repeated simulations, and obtain a value for \(E(X)\) using the algorithm in a programming language (e.g. Python). Find an approximate 95% confidence interval using the simulated data and verify the specified distribution by comparing our derived pdf of \(X\) and the given distribution. Lastly, check if the true value is within the computed 95% confidence interval.

Step by step solution

01

Set up the algorithm

Use Theorem 11.4.1 to mimic a procedure for generating observations with pdf \(f_{X}(x)\). Start by deriving \(Y\) from its given \(\Gamma(1,1)\) distribution. Then, simulate \(X\) given \(Y\) from its conditional pdf. A Python code for this algorithm would involve libraries such as numpy and scipy.
02

Estimation

To estimate \(E(X)\), the mean, you can repeatedly simulate the system designed in step 1 for a large number of times and then take an average of the resulting values of \(X\).
03

Estimate \(E(X)\) using the model

This step is about coding of step 2 model on a program. Python would be the preferred language. Create a script to simulate \(X\) (as per established in step 1 and 2) a large number of times (like 2000 times) and calculate a mean estimate using these simulations. This mean estimate will be the estimated value of \(E(X)\).
04

Confidence Interval

With a set of simulations from step 3, compute the 95% confidence interval for the mean estimated in step 3. This is typically done by calculating the standard deviation of simulated \(X\) values, dividing that by the square root of the number of simulations, and multiplying by the z-score corresponding to 95% confidence interval (about 1.96). Subtract and add this value from/to the mean for the interval.
05

Verify the distribution

Lastly, to verify that \(X\) has \(\Gamma(2,1)\) distribution, one needs to show that the pdf of \(X\) obtained from step 1 is equivalent to the pdf of a \(\Gamma(2,1)\) distribution. This might require integration and exchange of order of integration.
06

Compare with true value

Check whether the true value (which is '2') falls within the computed 95% confidence interval from step 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional probability density function
To comprehend the core concept of a conditional probability density function (pdf), imagine that we are dealing with two random variables, in our case, these are denoted as X and Y. The conditional pdf defines the probability of X occurring given that another event, Y, has already occurred. Given the conditional pdf,

\[\begin{equation}f(x \textbar y) = \begin{cases} e^{-(x-y)} & 0we can infer the likelihood of X assuming that Y has a certain value. The conditional pdf is essential for simulating the behavior of X under the constraint of Y. In educational contexts, it is vital to understand the concept of a conditional pdf to analyze the dependency between variables and predict outcomes in probability and statistics.
Expectation estimation
The expectation of a random variable gives us a long-term average and is a fundamental concept in statistics. It is denoted as E(X), which signifies the expected value of the random variable X. Estimating this expectation can be achieved through simulation when the direct calculation is complex or infeasible.

To estimate E(X) from simulations, one would run numerous experiments or simulations to generate a sequence of values of X and then compute their average. This process leverages the law of large numbers, which dictates that as the number of trials increases, the sample mean will tend to get closer to the expected value of the distribution. In our Gamma distribution simulation exercise, by simulating the variable X using the conditional pdf several times, we can estimate its expected value, thereby reinforcing the student's understanding of how expectation works in a practical, computational setting.
Monte Carlo simulation
Monte Carlo simulation is a powerful technique used to understand the impact of risk and uncertainty in prediction and forecasting models. The term 'Monte Carlo' refers to the Monte Carlo casino in Monaco due to the element of chance inherent in these stochastic (random) simulations. This simulation method utilizes random sampling to approximate complex mathematical or physical phenomena which are difficult or impossible to solve analytically.

In our gamma distribution simulation exercise, we employ Monte Carlo simulation to create a sequence of independent and identically distributed (iid) observations based on a given probability distribution (f_X(x)). By simulating thousands of outcomes using the defined conditional probability model, we collect data to estimate the expected value E(X) and calculate confidence intervals. This hands-on approach not only aids in conceptual understanding but also enhances computation skills, aligning perfectly with the educational objectives of making complex concepts accessible to students.

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Most popular questions from this chapter

Write the Bayes model of Exercise \(11.5 .2\) as $$ \begin{aligned} Y & \sim b(n, p), \quad 00 \end{aligned} $$ Set up the estimating equations for the mle of \(g(y \mid \theta)\), i.e., the first step to obtain the empirical Bayes estimator of \(p\). Simplify as much as possible.

Let \(Y_{n}\) be the \(n\) th order statistic of a random sample of size \(n\) from a distribution with pdf \(f(x \mid \theta)=1 / \theta, 00, \beta>0\). Find the Bayes solution \(\delta\left(y_{n}\right)\) for a point estimate of \(\theta\).

Let \(\mathbf{X}_{1}, \mathbf{X}_{2}, \ldots, \mathbf{X}_{n}\) be a random sample from a multivariate normal normal distribution with mean vector \(\boldsymbol{\mu}=\left(\mu_{1}, \mu_{2}, \ldots, \mu_{k}\right)^{\prime}\) and known positive definite covariance matrix \(\boldsymbol{\Sigma}\). Let \(\overline{\mathbf{X}}\) be the mean vector of the random sample. Suppose that \(\boldsymbol{\mu}\) has a prior multivariate normal distribution with mean \(\boldsymbol{\mu}_{0}\) and positive definite covariance matrix \(\boldsymbol{\Sigma}_{0}\). Find the posterior distribution of \(\mu\), given \(\overline{\mathbf{X}}=\overline{\mathrm{x}}\) Then find the Bayes estimate \(E(\boldsymbol{\mu} \mid \overline{\mathbf{X}}=\overline{\mathbf{x}})\).

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 \(\theta=0.5\) ?

Consider the following mixed discrete-continuous pdf for a random vector \((X, Y)\), (discussed in Casella and George, 1992): $$ f(x, y) \propto\left\\{\begin{array}{ll} \left(\begin{array}{l} n \\ x \end{array}\right) y^{x+\alpha-1}(1-y)^{n-x+\beta-1} & x=0,1, \ldots, n, 00\) and \(\beta>0\). (a) Show that this function is indeed a joint, mixed discrete-continuous pdf by finding the proper constant of proportionality. (b) Determine the conditional pdfs \(f(x \mid y)\) and \(f(y \mid x)\). (c) Write the Gibbs sampler algorithm to generate random samples on \(X\) and \(Y\). (d) Determine the marginal distributions of \(X\) and \(Y\). 11.4.8. If computation facilities are available, write a program for the Gibbs sampler of Exercise 11.4.7. Run your program for \(\alpha=10, \beta=4, m=3000\), and \(n=6000\). Obtain estimates (and confidence intervals) of \(E(X)\) and \(E(Y)\) and compare them with the true parameters.
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