Given current technology, the production of active matrix color screens for notebook computers is a difficult process that results in a significant proportion of defective screens being produced. At one company the daily proportion of defective \(9.5^{\prime \prime}\) and \(10.4^{\prime \prime}\) screens is the outcome of a bivariate random variable, \(X\), with joint density function \(f\left(x_{1}, X_{2} ; \alphaight)=\left(\alpha x_{1}+(2-\alpha) x_{2}ight) I_{[0,1]}\left(x_{1}ight) I_{[0,1]}\left(x_{2}ight)\), where \(\alpha \in(0,2)\). The daily proportions of defectives are independent from day to day. A collection of \(n\) iid outcomes of \(X\) will be used to generate an estimate of the \((2 \times 1)\) vector of mean daily proportions of defectives, \(\mu\), for the two types of screens being produced, as \(\underset{(2 \times 1)}{\overline{\mathbf{X}}_{n}}=\sum_{i=1}^{n} \mathbf{X}_{(i)} / n\), where \(\mathbf{X}_{(i)}=\left[\begin{array}{l}X_{1 i} \\ X_{2 i}\end{array}ight]\).

(a) Does \(\overline{\mathbf{X}}_{n} \xrightarrow{\text { as }} \boldsymbol{\mu}\) ? Does \(\overline{\mathbf{X}}_{n} \xrightarrow{p} \boldsymbol{\mu}\) ? Does \(\overline{\mathbf{X}}_{n} \xrightarrow{\text { d }} \boldsymbol{\mu}\) ?

(b) Define an asymptotic distribution for the bivariate random variable \(\overline{\mathbf{X}}_{n}\). If \(\alpha=1\) and \(n=200\), what is the approximate probability that \(\bar{X}_{n}[1]>.70\), given that \(\bar{X}_{n}[2]=.60\) ?

(c) Consider using an outcome of the function \(g\left(\overline{\mathbf{X}}_{n}ight)=\bar{X}_{n}[1] / \bar{X}_{n}[2]\) to generate an estimate of the relative expected proportions of defective \(9.5^{\prime \prime}\) and 10.4" screens, \(\mu_{1} / \mu_{2}\). Does \(g\left(\overline{\mathbf{X}}_{n}ight) \xrightarrow{\text { as }} \mu_{1} / \mu_{2}\) ? Does \(g\left(\overline{\mathbf{X}}_{n}ight) \xrightarrow{\mathrm{p}} \mu_{1} / \mu_{2}\) ? Does \(g\left(\overline{\mathbf{X}}_{n}ight) \xrightarrow{\mathrm{d}} \mu_{1} / \mu_{2}\) ?

(d) Define an asymptotic distribution for \(g\left(\overline{\mathbf{X}}_{n}ight)\). If \(\alpha=1\) and \(n=200\), what is the approximate probability that the outcome of \(g\left(\overline{\mathbf{X}}_{n}ight)\) will exceed 1?