Suppose that Xj ~ Poisson(A). Find the likelihood ratio statistic for testing H0: A = X vs. HA: A = 41, where A > 0. Show that the p-value for the likelihood ratio test can be calculated as P(CR1X > C1_I,A = X0). To do this, you will need to express the p-value calculation based on the likelihood ratio statistic in terms of C1_X and C1_T, and simplify. Be careful about switching the direction of your inequalities when you multiply by negative numbers, and note that log(A0/A1) < 0 since A0 < A1.
In probability, a fact you may have seen is that if X1, X2, ..., Xn are independent Poisson(A) random variables and we define Y to be the sum of the Xi's from i = 1 to n, then Y ~ Poisson(nA). Suppose A = 10. Use this to determine the rejection region for the test with a significance level in terms of y = C1_IT (for which values of y would you reject the null hypothesis?). You will want to use one of the R functions for the Poisson distribution.
What is the probability of Type I Error for the test you developed in part (c) if the null hypothesis is true? You will want to use one of the R functions for the Poisson distribution.
Now suppose that A0 = 10 and A1 = 20. What is the power of the test you developed in part (c)? You will want to use one of the R functions for the Poisson distribution.