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Problem 31

The number of births in randomly chosen hours of a day is as follows. $$ 4,0,6,5,2,1,2,0,4,3 . $$ Use this data to estimate the proportion of hours that had two or fewer births.

Expert verified

The proportion of hours that had two or fewer births is \(60\% \). To get this, 6 hours (the number of hours with two or fewer births) were divided by the total number of hours (10), resulting in 0.6. This was then multiplied by 100 to be expressed as a percentage.

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Chapter 4

Consider the normal distribution \(N\left(0, \sigma^{2}\right)\). With a random sample \(X_{1}, X_{2}, \ldots\), \(X_{n}\), we want to estimate the standard deviation \(\sigma .\) Find the constant \(c\) so that $Y=c \sum_{i=1}^{n}\left|X_{i}\right|\( is an unbiased estimator of \)\sigma$ and determine its efficiency.

Chapter 4

Prove that for the family of uniform distribution on the interval $[0, \theta]\(, \)\max \left(X_{1}, X_{2}, \ldots, X_{n}\right)$ is the MLE for \(\theta\).

Chapter 4

If \(X\) and \(Y\) are independent random variables with the same distribution and \(X+Y, X-Y\) are independent. Show that all random variables \(X, Y, X+Y, X-Y\) are normally distributed.

Chapter 4

Consider repeated observation on a \(m\) -dimensional random variable with mean $E\left(X_{i}\right)=\mu, i=1,2, \ldots, m, \quad \operatorname{Var}\left(X_{i}\right)=\sigma^{2}, i=1,2, \ldots, m$ and \(\operatorname{Cov}\left(X_{i}, X_{j}\right)=\rho \sigma^{2}, i \neq j .\) Let the \(i\) th observation be \(\left(x_{1 i}, \ldots, x_{m i}\right)\) \(i=1,2, \ldots, n\). Define $$ \begin{array}{c} \bar{X}_{i}=\frac{1}{m} \sum_{j=1}^{m} X_{j i} \\ W_{i}=\sum_{j=1}^{m}\left(X_{j i}-\bar{X}_{i}\right)^{2}, \\ B=m \sum_{i=1}^{n}\left(\bar{X}_{i}-\bar{X}\right)^{2}, \\ W=W_{1}+\cdots+W_{n} . \end{array} $$ where \(B\) is sum of squares between and \(W\) is sum of squares within samples. 1\. Prove (i) \(\left.W \sim(1-\rho) \sigma^{2} \chi^{(} m n-n\right)\) and (ii) \(B \sim(1+(m-1) \rho) \sigma^{2} \chi^{2}(n-1)\). 2\. Suppose \(\frac{(1-\rho) B}{(1+(m-1) \rho) W} \sim F_{(n-1),(m n-n)} .\) Prove that when \(\rho=0, \frac{W}{W+B}\) follows beta distribution with parameters \(\frac{m n-n}{2}\) and \(\frac{n-1}{2}\).

Chapter 4

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from uniform distribution on an interval \((0, \theta)\). Show that $\left(\prod_{i=1}^{n} X_{i}\right)^{1 / n}\( is consistent estimator of \)\theta e^{-1}$.

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