$2 . \mathrm{I} . 3 \mathrm{H} \quad$

Explain what is meant by a uniformly most powerful test, its power function and size.

Let $Y_{1}, \ldots, Y_{n}$ be independent identically distributed random variables with common density $\rho e^{-\rho y}, y \geq 0$. Obtain the uniformly most powerful test of $\rho=\rho_{0}$ against alternatives $\rho<\rho_{0}$ and determine the power function of the test.

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