***Please only answer Q2a

Exercise 4.1 (*ww - Likelihood principle and p-values (50%)). (Background) Suppose that, in a social media "FaceLook", advertisements may pop up to selected users accordingly to a deep-learning (DL) algorithm. If a selected user clicks into the advertisement, then we say that the algorithm successfully selects a correct user. Let @ e (0, 1) be the probability of a successful selection. Assume that a random algorithm (i.e., randomly pops up an advertisement to users) leads to a successful selection rate 0 = 15%. FaceLook would like to test the effectiveness of the DL algorithm by testing Ho : 0 = 15% against H1 : 0 > 15%. (4.1) Denote success and failures by 1 and 0, respectively. FaceLook obtains the following dataset 010 0 0 0 01 10 10 010 1. (Experiment 1) Suppose that the algorithm selects 15 users. It records 5 successes. Then it is reasonable to model the number of successes by X - Bin(15, #), whose PMF is fo = (15)07(1 - 0)15-21(x =0, . .., 15). (a) (10%) Derive the likelihood function L(0 | X). (b) (10%) Explain why the p-value for testing (4.1) is p = >res fos(I). Is Ho rejected at 5% level? 2. (Experiment 2) Suppose that the algorithm repeatedly selects users until 10 failures. It takes 15 trials. Then it is reasonable to model the number of trials needed by N ~ 10 + NB(10, 0), whose PMF is fo(n) = (10_1)92-10(1 -0) 101(n = 10, 11, ...). (a) (10%) Derive the likelihood function L(0 | N). (b) (10%) Explain why the p-value for testing (4.1) is p = En-is fo.15(n). Is Ho rejected at 5% level?Example 5.3 is related. . p-value is the probability that the data are at least as extreme (in the direction of H, ) as the actually realized observations under Ho. We reject Ho at 5% level if the p-value is less than 5%.Example 5.3 (www Proportional likelihood). 1. Let X ~ Bin(4, 0), where o e (0, 1). Suppose X = z is realized. Then the likelihood of d is L(0 | x) = fo(x) = = (x )07(1-0)" -1(1=0,..., 4). In particular, if r = 3, then L(0 | 3) = (5.2) 2. Let N ~ 3 + NB(3, 1 -8), where o c (0, 1). Suppose N = n is realized. Then the likelihood of d is In particular, if n = 4, then L(0 | 4) = 4 03 (1 - 0)4-3 xx 03( 1 - 0). (5.3) Hence, the likelihoods (5.2) and (5.3) are proportional to each other