91Ó°ÊÓ

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a Poisson distribution with parameter \(\theta>0 .\) From the Remark of this section, we know that \(E\left[(-1)^{x_{1}}\right]=e^{-2 \theta}\). (a) Show that \(E\left[(-1)^{x_{1}} \mid Y_{1}=y_{1}\right]=(1-2 / n)^{y_{1}}\), where \(Y_{1}=X_{1}+X_{2}\) \(+\cdots+X_{n} .\) Hint. First show that the conditional p.d.f. of \(X_{1}, X_{2}, \ldots, X_{n-1}\) given \(Y_{1}=y_{1}\), is multinomial, and hence that of \(X_{1}\) given \(Y_{1}=y_{1}\) is \(b\left(y_{1}, 1 / n\right)\) (b) Show that the maximum likelihood estimator of \(e^{-2 \theta}\) is \(e^{-2 \bar{X}}\). (c) Since \(y_{1}=n \bar{x}\), show that \((1-2 / n)^{y_{1}}\) is approximately equal to \(e^{-2 \bar{x}}\) when \(n\) is large.

Let \(Y_{1}

Let \(\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)\) denote a random sample of size \(n\) from a bivariate normal distribution with means \(\mu_{1}\) and \(\mu_{2}\), positive variances \(\sigma_{1}^{2}\) and \(a_{2}^{2}\), and correlation coefficient \(\rho .\) Show that \(\sum_{1}^{n} X_{i}, \sum_{1}^{n} Y_{i}\) \(\sum_{1}^{n} X_{i}^{2}, \sum_{1}^{n} Y_{i}^{2}\), and \(\sum_{1}^{\pi} X_{i} Y_{i}\) are joint sufficient statistics for the five parameters. Are \(X=\sum_{1}^{n} X_{i} / n, Y=\sum_{1}^{n} Y_{i} / n, S_{1}^{2}=\sum_{1}^{n}\left(X_{i}-X\right)^{2} / n, S_{2}^{2}=\sum_{1}^{n}\left(Y_{i}-\nabla\right)^{2} / n\) and \(\sum_{1}^{n}\left(X_{i}-X\right)\left(Y_{i}-\nabla\right) / n S_{1} S_{2}\) also joint sufficient statistics for these parameters?

Let the p.d.f. \(f\left(x ; \theta_{1}, \theta_{2}\right)\) be of the form $$ \exp \left[p_{1}\left(\theta_{1}, \theta_{2}\right) K_{1}(x)+p_{2}\left(\theta_{1}, \theta_{2}\right) K_{2}(x)+S(x)+q\left(\theta_{1}, \theta_{2}\right)\right], \quad a

Let \(Y_{1}