Question

Let $X$ be a single observation drawn from $U(0, \theta)$ distribution, where $\theta \in \{1, 2\}$. To test $H_0: \theta = 1$ against $H_1: \theta = 2$, consider the test procedure that rejects $H_0$ if and only if $X > 0.75$. If the probabilities of Type-I and Type-II errors are $\alpha$ and $\beta$, respectively, then $\alpha + \beta$ equals ......... (round off to two decimal places).
 

Hint:

For uniform distributions, probabilities are proportional to interval lengths. Use geometry of the support directly to find $\alpha$ and $\beta$.

Answer 1

Correct Answer: 0.61

Step 1: Recall Type-I and Type-II error definitions.
Type-I error ($\alpha$): Reject $H_0$ when $H_0$ is true. Type-II error ($\beta$): Fail to reject $H_0$ when $H_1$ is true.

Step 2: Under $H_0: \theta = 1$.
$X \sim U(0, 1)$. \[ \alpha = P(X > 0.75 | \theta = 1) = 1 - 0.75 = 0.25. \]

Step 3: Under $H_1: \theta = 2$.
$X \sim U(0, 2)$. \[ \beta = P(X \le 0.75 | \theta = 2) = \frac{0.75 - 0}{2} = 0.375. \]

Step 4: Compute total error probability.
\[ \alpha + \beta = 0.25 + 0.375 = 0.625. \]

Step 5: Round off.
\[ \boxed{\alpha + \beta = 0.62.} \]