/*! 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} Problem 1 Let \(X_{1}, X_{2}\), and \(X_{3... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 1

Let \(X_{1}, X_{2}\), and \(X_{3}\) have a multinomial distribution in which \(n=25, k=4\), and the unknown probabilities are \(\theta_{1}, \theta_{2}\), and \(\theta_{3}\), respectively. Here we can, for convenience, let \(X_{4}=25-X_{1}-X_{2}-X_{3}\) and \(\theta_{4}=1-\theta_{1}-\theta_{2}-\theta_{3} .\) If the observed values of the random variables are \(x_{1}=4, x_{2}=11\), and \(x_{3}=7\), find the maximum likelihood estimates of \(\theta_{1}, \theta_{2}\), and \(\theta_{3}\).

Short Answer

Expert verified
The maximum likelihood estimates of \(\theta_{1}, \theta_{2}\), and \(\theta_{3}\) are: \(\hat{\theta_{1}} = 0.16\), \(\hat{\theta_{2}} = 0.44\), and \(\hat{\theta_{3}} = 0.28\).

Step by step solution

01

Understand the Properties of Multinomial Distribution

By definition, for a multinomial distribution, the probabilities sum to 1. Therefore, we have that \(\theta_{1} + \theta_{2} + \theta_{3} + \theta_{4} = 1\). Since \(X_{4}\) is presented as a constant, we can formulate that \(\theta_{4} = 1 - \theta_{1} - \theta_{2} - \theta_{3}\). The observed values are \(x_{1}=4, x_{2}=11\), and \(x_{3}=7\), and the total number of trials is \(n = 25\). This means \(x_{4} = n - x_{1} - x_{2} - x_{3} = 25 - 4 - 11 - 7 = 3\).
02

Calculate the Maximum Likelihood Estimates

Under a multinomial distribution model, the maximum likelihood estimates are determined by the observed data. They are obtained by dividing the observed values by the total number of trials: \(\hat{\theta_{i}} = x_{i} / n\), where \(i\) represents the category or class. So here, the maximum likelihood estimates of \(\theta_{1}, \theta_{2}\), and \(\theta_{3}\) are \(\hat{\theta_{1}} = x_{1} / n = 4 / 25 = 0.16\), \(\hat{\theta_{2}} = x_{2} / n = 11 / 25 = 0.44\) and \(\hat{\theta_{3}} = x_{3} / n = 7 / 25 = 0.28\). And \(\hat{\theta_{4}} = x_{4} / n = 3 / 25 = 0.12\).

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Most popular questions from this chapter

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