Let Y1 , Y2, . . . , Yn denote a random sample from a

Chapter 9, Problem 38E

(choose chapter or problem)

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample from a normal distribution with mean \(\mu\) and variance \(\sigma^{2}\)

a. If \(\mu\) is unknown and \(\sigma^{2}\) is known, show that \(\bar{Y}\) is sufficient for \(\mu\).

b. If \(\mu\) is known and \(\sigma^{2}\) is unknown, show that \(\sum_{i=1}^{n}\left(Y_{i}-\mu\right)^{2}\) is sufficient for \(\sigma^{2}\).

c. If \(\mu\) and \(\sigma^{2}\) are both unknown, show that \(\sum_{i=1}^{n} Y_{i}\) and \(\sum_{i=1}^{n} Y_{i}^{2}\) are jointly sufficient for \(\mu\) and \(\sigma^{2}\). [Thus, it follows that \(\bar{Y}\) and \(\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}\) or \(\bar{Y}\) and \(S^{2}\) are also jointly sufficient for \(\mu\) and \(\sigma^{2}\).]

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