91Ó°ÊÓ

Suppose a random sample of size \(n\) is drawn from the probability model $$ p_{X}(k ; \theta)=\frac{\theta^{2 k} e^{-9^{2}}}{k !}, \quad k=0,1,2, \ldots $$ Find a formula for the maximum likelihood estimator, \(\hat{\theta}\).

Suppose a random sample of size \(n\) is drawn from a normal pdf where the mean \(\mu\) is known but the variance \(\sigma^{2}\) is unknown. Use the method of maximum likelihood to find a formula for \(\theta^{2}\). Compare your answer to the maximum likelihood estimator found in Example \(5.2 .5 .\)

. Use the method of moments to estimate \(\theta\) in the pdf $$ f_{Y}(y ; \theta)=\left(\theta^{2}+\theta\right) y^{\theta-1}(1-y), \quad 0 \leq y \leq 1 $$ Assume that a random sample of size \(n\) has been collected.

A criminologist is searching through FBI files to document the prevalence of a rare double-whorl fingerprint. Among six consecutive sets of 100,000 prints scanned by a computer, the numbers of persons having the abnormality are \(3,0,3,4,2\), and 1, respectively. Assume that double whorls are Poisson events. Use the method of moments to estimate their occurrence rate, \(\lambda\). How would your answer change if \(\lambda\) were estimated using the method of maximum likelihood?

A commonly used IQ test is scaled to have a mean of 100 and a standard deviation of \(\sigma=15\). A school counselor was curious about the average IQ of the students in her school and took a random sample of fifty students' IQ scores. The average of these was \(\bar{y}=107.9\). Find a \(95 \%\) confidence interval for the student \(\mathrm{IQ}\) in the school.

University officials are planning to audit 1586 new appointments to estimate the proportion \(p\) who have been incorrectly processed by the payroll department. (a) How large does the sample size need to be in order for \(\frac{x}{n}\), the sample proportion, to have an \(85 \%\) chance of lying within \(0.03\) of \(p\) ? (b) Past audits suggest that \(p\) will not be larger than \(0.10\). Using that information, recalculate the sample size asked for in part (a).

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample of size \(n\) drawn from a Poisson pdf where \(\lambda\) is an unknown parameter. Show that \(\hat{\lambda}=\bar{X}\) is unbiased for \(\lambda\). For what type of parameter, in general, will the sample mean necessarily be an unbiased estimator? (Hint: The answer is implicit in the derivation showing that \(\bar{X}\) is unbiased for the Poisson \(\lambda\).)

Let \(Y_{\min }\) be the smallest order statistic in a random sample of size \(n\) drawn from the uniform pdf, \(f_{Y}(y ; \theta)=\) \(1 / \theta, 0 \leq y \leq \theta\). Find an unbiased estimator for \(\theta\) based on \(Y_{\min }\).

Suppose that \(14,10,18\), and 21 constitute a random sample of size 4 drawn from a uniform pdf defined over the interval \([0, \theta]\), where \(\theta\) is unknown. Find an unbiased estimator for \(\theta\) based on \(Y_{3}^{\prime}\), the third order statistic. What numerical value does the estimator have for these particular observations? Is it possible that we would know that an estimate for \(\theta\) based on \(Y_{3}^{\prime}\) was incorrect, even if we had no idea what the true value of \(\theta\) might be? Explain.

A sample of size 1 is drawn from the uniform pdf defined over the interval \([0, \theta]\). Find an unbiased estimator for \(\theta^{2}\). (Hint: Is \(\theta=Y^{2}\) unbiased?)