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 book, and 4 receives 1’s book. This outcome can be abbreviated (2,4,3,1).

Let X denote the number of students who receive their own book. Determine the pmf of X.

Question

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 book, and 4 receives 1’s book. This outcome can be abbreviated (2,4,3,1).

Let X denote the number of students who receive their own book. Determine the pmf of X.

ybarrap · Answer

Let Ni = Probability of No matches for ith student Mi = Probability of Match for ith student P[X = 1] = P[1 match] = 4C1 * P[M1]*P[N2|M1]*P[N3|M1,N2]*P[N4|M1,N2,N3] = 4!/(1!3!) * 1/4 * 2/3 * 1/2 * 1 = 8/24 P[X=2] = P[2 matches] = 4C2 * P[M1]*P[M2|M1]*P[N3|M1,M2]*P[N4|M1,M2,N3] = 4!/(2!2!) * 1/4 * 1/3 * 1/2 * 1 = 6/24 P[X=3] = P[3 matches] = 4C3 * P[M1]*P[M2|M1]*P[M3|M1,M2]*P[N4|M1,M2,M3] = 0, since P[N4|M1,M2,M3]=0 P[X=4] = P[4 matches] = 4C4 * P[M1]*P[M2|M1]*P[M3|M1,M2]*P[M4|M1,M2,M3] = 1 * 1/4 * 1/3 * 1/2 * 1 = 1/24 P[X=0] = P[0 matches] = 1 - P[1 matches] - P[2 matches] - P[3 matches] - P[4 matches] = 1 - 8/24 - 6/24 - 1/24 = 9/24 Ans. P[X=0] = 9/24, P[X=1] = 8/24,P[X=2]=6/24,P[X=3]=0,P[X=4]=1/24 Here's a plot: http://www.wolframalpha.com/input/?i=plot+%28%280%2C9%2F24%29%2C%281%2C8%2F24%29%2C%282%2C6%2F24%29%2C%283%2C0%2F24%29%2C%284%2C1%2F24%29%29