3. Indicator and indicator-like random variables [7 points each]

(a) 21 people enter a large elevator at the basement of a building with 40 floors above.

Assume that there are no other persons in the building. Let X denote the number of

stops that the elevator makes at which exactly 3 people exit. Find E[X].

Example: Suppose 13 people exit on floor 2, 3 people exit on floor 10, 2 people exit on

floor 26 and 3 people exit on floor 35, then X = 2.

(b) A fair coin, with outcomes 0 and 1, is tossed repeatedly M times. Let X denote the

number of runs of exactly 2 ones. Find E[X].

Example: Let M = 16, and let the coin toss realization be 0100110110111011. Then

X = 3.

(c) K numbers are chosen randomly from numbers {1, 2, . . . , N} uniformly without replace

ment, where K . Let X denote the smallest of the K numbers chosen. Find E[X].

(d) Five boys and five girls go to a dinner party. They are seated at random around a circular

table with 10 chairs. Let X denote the number of girls sitting next to a boy. Find E(X).

3. Indicator and indicator-like random variables [7 points each] (a) 21 people enter a large elevator at the basement of a building with 40 floors above. Assume that there are no other persons in the building. Let X denote the number of stops that the elevator makes at which exactly 3 people exit. Find E[X]. Example: Suppose 13 people exit on floor 2, 3 people exit on floor 10, 2 people exit on floor 26 and 3 people exit on floor 35, then X = 2. (b) A fair coin, with outcomes 0 and 1, is tossed repeatedly M times. Let X denote the number of runs of exactly 2 ones. Find E[X]. Example: Let M = 16, and let the coin toss realization be 0100110110111011. Then X = 3. (c) K numbers are chosen randomly from numbers {1, 2, ..., N} uniformly without replace- ment, where K