After all students have left the classroom, a statistics professor notices that four copies of the text were left under desks. At the beginning of the next lecture, the professor distributes the four books in a completely random fashion to each of the four students (1, 2, 3, and 4) who claim to have left books. One possible outcome is that 1 receives 2’s book, 2 receives 4’s book, 3 receives his or her own book, and 4 receives 1’s book. This outcome can be abbreviated as (2, 4, 3, 1). a.? ?List the other 23 possible outcomes. b. ?Let X denote the number of students who receive their own book. Determine the pmf of X.

Answer: Step 1 of 2 At the beginning of the next lecture, the professor distributes the four books in a completely random fashion to each of the four students (1, 2, 3, and 4) who claim to have left books. One possible outcome is that 1 receives 2’s book, 2 receives 4’s book, 3 receives his or her own book, and 4 receives 1’s book. Let the string (2, 4, 3, 1) denote the order that four textbooks are handed back to four students who claim to be missing their books. Let Y be the number of students who get their book back when the books are handed back in a random order. The 24 possible outcomes and corresponding Y values are: The probability distribution or probability mass function (pmf) of a discrete random variable is defined for every number by P(x) = P(X = x) = P(all sS : X (s) = x ) P(X = x) is read “the probability that the random variable X assumes the value x”. The Outcomes for experiment described as Possible y Possible y Possible y Possible y 1234 4 2134 2 3124 1 4123 0 1243 2 2143 0 3142 0 4132 1 1432 2 2413 0 3412 0 4213 1 1342 1 2431 1 3421 0 4231 2 1423 1 2314 1 3241 1 4312 0 4324 2 2341 0 3214 1 4321 0