Q1: The Association of Chefs Societies (ACS) recruit reviewers every year to evaluate chef's cooking skills. The reviewer candidate must pass a consistency test before receiving the offer. ACS provides 4 similarly-looking dishes cooked by 4 experienced chefs following the same protocol. The reviewer candidate will taste the series of dishes twice and give a rank from the best to worst. Suppose the four dishes were ranked 1, 2, 3, 4 in the first round of ranking, but ranked a1, a2, a3, a4 in the second round of ranking. ACS makes the decision through the score X, as given below. Lower X score indicates better consistency.

X=|1-a1|+|2-a2|+|3-a3|+|4-a4|

1. If the ranking is completely random, write down the PMF for X.
2. If a reviewer candidate randomly ranks all the dishes, what is E(X)?

There are 2 rounds in ACS test. Candidates have X1 and X2 as their scores, and the overall X is given as X=X1+X2. If candidates get a high X score that happens less than 5% of the times in all random ranking procedures, then they pass. What is the minimum X score for passing?

Q1: The Association of Chefs Societies (ACS) recruit reviewers every year to evaluate chef's cooking skills. The reviewer candidate must pass a consistency test before receiving the offer. ACS provides 4 similarly-looking dishes cooked by 4 experienced chefs following the same protocol. The reviewer candidate will taste the series of dishes twice and give a rank from the best to worst. Suppose the four dishes were ranked 1, 2, 3, 4 in the rst round of ranking, but ranked a1, 32, a3, a1 in the second round of ranking. ACS makes the decision through the score X, as given below. Lower X score indicates better consistency. X= |1 a'1|+|2a2|+l3a3|+I4a4| 1. If the ranking is completely random, write down the PMF forX. 2. If a reviewer candidate randomly ranks all the dishes, what is E(X)? 3. There are 2 rounds in ACS test. Candidates have X1 and X2 as their scores, and the overall X is given as X = XI + X2 . If candidates get a highX score that happens less than 5% of the times in all random ranking procedures, then they pass. What is the minimum X score for passing