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.

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.

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 \(\theta\).

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.

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 \(\frac{n-1}{2}\).

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}$.