A statistical variable X can have only one of the two following distributions:
H0: f0(x) = 0.2-0.02x
H1: f1(x) = 0.02x
In both cases, the support of the statistical variable is:
0 <= X <= 10
The decision rule for to test the hypothesis, based on a single casual observation, consider the following rejection region for H0:
Rif = {X >= 8}
Is is required to:
a) Evaluate the significance level (alpha);
b) Evaluate the power of the test (1-beta);
My partial solution: I have written that alpha is the probability to reject H0 when H0 is true (error of first type). 1-beta is the probability to reject H0 when H0 is false (this means, the probability to make the correct decision).
How can I continue to solve?
Thank you very much for considering my request.
If $H_0$ is true, then $X$ has the probability density $f_0(x) = 0.2 - 0.02x$ for $0 \le x \le 10$. So what is $\Pr[X \ge 8]$ under this assumption? It is $$\Pr[X \ge 8] = \int_{x=8}^{10} f_0(x) \, dx = \ldots?$$
Similarly, under the assumption that $H_1$ is true, what is the probability $\Pr[X \ge 8]$?