Problem 6
Construct a rejection algorithm to simulate from \(f(x)=30 x(1-x)^{4}, 0 \leq x \leq 1\), using the \(U(0,1)\) density as the proposal function \(g\). Give its efficiency.
Problem 8
If \(R\) is binomial with denominator \(m\) and probability \(\pi\), show that $$ \frac{R / m-\pi}{\\{\pi(1-\pi) / m\\}^{1 / 2}} \stackrel{D}{\longrightarrow} Z \sim N(0,1) $$ and that the limits of a \((1-2 \alpha)\) confidence interval for \(\pi\) are the solutions to $$ R^{2}-\left(2 m R+m z_{\alpha}^{2}\right) \pi+m\left(m+z_{\alpha}^{2}\right) \pi^{2}=0 $$ Give expressions for them. In a sample with \(m=100\) and 20 positive responses, the \(0.95\) confidence interval is \((0.13,0.29)\). As this interval either does or does not contain the true \(\pi\), what is the meaning of the \(0.95 ?\)
Problem 8
Let \(R_{1}, R_{2}\) be independent binomial random variables with probabilities \(\pi_{1}, \pi_{2}\) and denominators \(m_{1}, m_{2}\), and let \(P_{i}=R_{i} / m_{i} .\) It is desired to test if \(\pi_{1}=\pi_{2}\). Let \(\widehat{\pi}=\left(m_{1} P_{1}+m_{2} P_{2}\right) /\left(m_{1}+m_{2}\right) .\) Show that when \(\pi_{1}=\pi_{2}\), the statistic $$ Z=\frac{P_{1}-P_{2}}{\sqrt{\widehat{\pi}(1-\hat{\pi})\left(1 / m_{1}+1 / m_{2}\right)}} \stackrel{D}{\longrightarrow} N(0,1) $$ when \(m_{1}, m_{2} \rightarrow \infty\) in such a way that \(m_{1} / m_{2} \rightarrow \xi\) for \(0<\xi<1\). Now consider a \(2 \times 2\) table formed using two independent binomial variables and having entries \(R_{i}, S_{i}\) where \(R_{i}+S_{i}=m_{i}, R_{i} / m_{i}=P_{i}\), for \(i=1,2\). Show that if \(\pi_{1}=\pi_{2}\) and \(m_{1}, m_{2} \rightarrow \infty\), then $$ X^{2}=\left(n_{1}+n_{2}\right)\left(R_{1} S_{2}-R_{2} S_{1}\right)^{2} /\left\\{n_{1} n_{2}\left(R_{1}+R_{2}\right)\left(S_{1}+S_{2}\right)\right\\} \stackrel{D}{\longrightarrow} \chi_{1}^{2} $$ Two batches of trees were planted in a park: 250 were obtained from nursery \(A\) and 250 from nursery \(B\). Subsequently 41 and 64 trees from the two groups die. Do trees from the two nurseries have the same survival probabilities? Are the assumptions you make reasonable?
Problem 9
If \(Z \sim N(0,1)\), derive the density of \(Y=Z^{2}\). Although \(Y\) is determined by \(Z\), show they are uncorrelated.
Problem 9
If \(\bar{Y}\) is the average of a random sample \(Y_{1}, \ldots, Y_{n}\) from density \(\theta^{-1} \exp (-y / \theta), y>0\) \(\theta>0\), give the limiting distribution of \(Z(\theta)=n^{1 / 2}(\bar{Y}-\theta) / \theta\) as \(n \rightarrow \infty .\) Hence obtain an approximate two-sided \(95 \%\) confidence interval for \(\theta\). Show that for large \(n, \log (\bar{Y}) \doteq \log \theta+n^{-1 / 2} Z\), find an approximate mean and variance for \(\log \bar{Y}\), and hence give another approximate two-sided \(95 \%\) confidence interval for \(\theta\). Which interval would you prefer in practice?
Problem 9
I am uncertain about what will happen when I next roll a die, about the exact amount of money at present in my bank account, about the weather tomorrow, and about what will happen when I die. Does uncertainty mean the same thing in all these contexts? For which is variation due to repeated sampling meaningful, do you think?
Problem 10
If \(W \sim \chi_{v}^{2}\), show that \(\mathrm{E}(W)=v, \operatorname{var}(W)=2 v\) and \((W-v) / \sqrt{2 v} \stackrel{D}{\longrightarrow} N(0,1)\) as \(v \rightarrow\) \(\infty\)
Problem 10
Independent pairs \(\left(X_{j}, Y_{j}\right), j=1, \ldots, m\) arise in such a way that \(X_{j}\) is normal with mean \(\lambda_{j}\) and \(Y_{j}\) is normal with mean \(\lambda_{j}+\psi, X_{j}\) and \(Y_{j}\) are independent, and each has variance \(\sigma^{2} .\) Find the joint distribution of \(Z_{1}, \ldots, Z_{m}\), where \(Z_{j}=Y_{j}-X_{j}\), and hence show that there is a \((1-2 \alpha)\) confidence interval for \(\psi\) of form \(A \pm m^{-1 / 2} B c\), where \(A\) and \(B\) are random variables and \(c\) is a constant. Obtain a \(0.95\) confidence interval for the mean difference \(\Psi\) given \((x, y)\) pairs \((27,26)\), \((34,30),(31,31),(30,32),(29,25),(38,35),(39,33),(42,32) .\) Is it plausible that \(\psi \neq 0 ?\)
Problem 11
(a) If \(F \sim F_{v_{1}, v_{2}}\), show that \(1 / F \sim F_{v_{2}, v_{1}}\). Give the quantiles of \(1 / F\) in terms of those of \(F\) (b) Show that as \(v_{2} \rightarrow \infty, v_{1} F\) tends in distribution to a chi-squared variable, and give its degrees of freedom. (c) If \(Y_{1}\) and \(Y_{2}\) are independent variables with density \(e^{-y}, y>0\), show that \(Y_{1} / Y_{2}\) has the \(F\) distribution, and give its degrees of freedom.
Problem 12
Let \(f(t)\) denote the probability density function of \(T \sim t_{v}\). (a) Use \(f(t)\) to check that \(\mathrm{E}(T)=0, \operatorname{var}(T)=v /(v-2)\), provided \(v>1,2\) respectively. (b) By considering \(\log f(t)\), show that as \(v \rightarrow \infty, f(t) \rightarrow \phi(t)\).
