Suppose (Y_{i}) is distributed i.i.d. (Nleft(0, sigma^{2} ight)) for (i=1,2, ldots, n). a. Show that (Eleft(Y_{i}^{2} /

Suppose \(Y_{i}\) is distributed i.i.d. \(N\left(0, \sigma^{2}\right)\) for \(i=1,2, \ldots, n\).

a. Show that \(E\left(Y_{i}^{2} / \sigma^{2}\right)=1\).

b. Show that \(W=\left(1 / \sigma^{2}\right) \sum_{i=1}^{n} Y_{i}^{2}\) is distributed \(\chi_{n}^{2}\).

c. Show that \(E(W)=n\). [Hint: Use your answer to (a).]

d. Show that \(V=Y_{1} / \sqrt{\frac{\sum_{i=2}^{n} Y_{i}^{2}}{n-1}}\) is distributed \(t_{n-1}\).