Question

Let \(X_1,X_2\) be a random sample of size \(2\) from a distribution with the probability density function

Hint:

For simple versus simple testing, the Neyman-Pearson lemma says to reject \(H_0\) for large values of the likelihood ratio \(\frac{L_1}{L_0}\).

Answer 1

Correct Answer: 0.615

Step 1: Shape of the most powerful test.
The likelihood ratio $\frac{L(2)}{L(1)}=4X_1X_2$ grows with $X_1X_2$, so reject for large $X_1X_2$.

Step 2: Observed statistic.
With $X_1=\frac14$, $X_2=\frac12$, $X_1X_2=\frac18$.

Step 3: Define the $p$-value.
Under $H_0$ ($\alpha=1$), $X_1,X_2\sim U(0,1)$, so $p=P(UV\ge\frac18)$.

Step 4: Evaluate.
Using $P(UV<c)=c(1-\ln c)$ with $c=\frac18$, the complement is $\frac18(1+\ln8)=0.3849$, so $p=1-0.3849=0.6151$.

Step 5: Round.
\[ \boxed{0.615} \]