I am studying about the discrete and the continuous variable random and here is one problem that requires me to write down PMF , and do some calculations.

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, 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: |l a1|+|2-a2|+|3a3|+|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)? 3. There are 2 rounds in ACS test. Candidates have X1 and X2 as their scores, and the overall X is given as X = X 1 + X 2 . 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