1. "A single observation of a random variable having a hypergeometric distribution with N = 7 and n = 2 is used to test the null hypothesis k = 2 against the alternative hypothesis k = 4. If the null hypothesis is rejected if and only if the value of the random variable is 2, find the probabilities of type I and II errors."
2. "A single observation of a random variable having a geometric distribution is used to test the null hypothesis \(\displaystyle \theta = \theta_0\) against the alternative hypothesis \(\displaystyle \theta > \theta_0\). If the null hypothesis is rejected if and only if the observed value of the random variable is greater than or equal to the positive integer k, find expressions for the probabilities of type I and II errors."
As usual, I am having a very tough time getting these problems started. For #1, the hypergeometric, how do I even get all the info just to set up the PDF? I.e.:
\(\displaystyle f(x) = \frac{{M \choose 2} {{7 - M} \choose {2-2}}} {7 \choose 2}\)
What is M?? It can't be 1, because 1 choose 2 is undefined. How do I even set this problem up?
And #2 isn't any better. I know that here, \(\displaystyle f(x) = \theta (1- \theta)^{x-1}\). But trying to integrate this from 1 to k-1 results in a complete mess. HELP!
2. "A single observation of a random variable having a geometric distribution is used to test the null hypothesis \(\displaystyle \theta = \theta_0\) against the alternative hypothesis \(\displaystyle \theta > \theta_0\). If the null hypothesis is rejected if and only if the observed value of the random variable is greater than or equal to the positive integer k, find expressions for the probabilities of type I and II errors."
As usual, I am having a very tough time getting these problems started. For #1, the hypergeometric, how do I even get all the info just to set up the PDF? I.e.:
\(\displaystyle f(x) = \frac{{M \choose 2} {{7 - M} \choose {2-2}}} {7 \choose 2}\)
What is M?? It can't be 1, because 1 choose 2 is undefined. How do I even set this problem up?
And #2 isn't any better. I know that here, \(\displaystyle f(x) = \theta (1- \theta)^{x-1}\). But trying to integrate this from 1 to k-1 results in a complete mess. HELP!