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

QUESTION:

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample from a population having a Poisson distribution with mean \(\lambda\).

a Find the form of the rejection region for a most powerful test of \(H_{0}: \lambda_{0}=\lambda_{0}\) against \(H_{a}: \lambda=\lambda_{2}\), where \(\lambda_{a}>\lambda_{0}\)

b Recall that \(\sum_{i=1}^{n} Y_{i}\) has a Poisson distribution with mean . Indicate how this information can be used to find any constants associated with the rejection region derived in part (a).

c Is the test derived in part (a) uniformly most powerful for testing \(H_{0}: \lambda=\lambda_{0}\) against \(H_{a}: \lambda=\lambda_{a}\)? Why?

d Find the form of the rejection region for a most powerful test of \(H_{0}: \lambda=\lambda_{0}\) against \(\(H_{a}: \lambda=\lambda_{a}\)\), where \(\lambda_{a}<\lambda_{0}\)

Equation transcription:

Text transcription:

Y_{1}, Y_{2}, \ldots, Y_{n}

\lambda

H_{0}: \lambda_{0}=\lambda_{0}

H_{a}: \lambda=\lambda_{2}

\lambda_{a}>\lambda_{0}

\sum_{i=1}^{n} Y_{i}

H_{0}: \lambda=\lambda_{0}

H_{a}: \lambda=\lambda_{a}

\lambda_{a}<\lambda_{0}