Suppose that \(Y_{1}, Y_{2}, \ldots, Y_{n}\) are random variables with a common mean \(\mu_{Y}\); a common variance \(\sigma_{Y}^{2}\); and the same correlation \(ho\) (so that the correlation between \(Y_{i}\) and \(Y_{j}\) is equal to \(ho\) for all pairs \(i\) and \(j\), where \(i eq j\) ).

a. Show that \(\operatorname{cov}\left(Y_{i}, Y_{j}\right)=ho \sigma_{Y}^{2}\) for \(i eq j\).

b. Suppose that \(n=2\). Show that \(E(\bar{Y})=\mu_{Y}\) and \(\operatorname{var}(\bar{Y})=\frac{1}{2} \sigma_{Y}^{2}+\frac{1}{2} ho \sigma_{Y}^{2}\).

c. For \(n \geq 2\), show that \(E(\bar{Y})=\mu_{Y}\) and \(\operatorname{var}(\bar{Y})=\sigma_{Y}^{2} / n+\) \([(n-1) / n] ho \sigma_{Y}^{2}\).

d. When \(n\) is very large, show that \(\operatorname{var}(\bar{Y}) \simeq ho \sigma_{Y}^{2}\).