18.5. Linearity of Expectation 781
Problem 18.18(Deviations from the mean).
LetBbe a random variable with unbiased binomial distribution, nemely,
PrŒBDkçDn
k!
2 n:Assumenis even. Prove the Lemma about the expected absolute deviation ofB
from its mean:
Lemma.
ExŒjB ExŒBçjçDn
n
2!
n
2 nC^1:
Problems for Section 18.5
Practice Problems
Problem 18.19.
MIT students sometimes delay doing laundry until they finish their problem sets.
Assume all random values described below are mutually independent.
(a)Abusystudent must complete 3 problem sets before doing laundry. Each
problem set requires 1 day with probability2=3and 2 days with probability1=3.
LetBbe the number of days a busy student delays laundry. What is ExŒBç?
Example: If the first problem set requires 1 day and the second and third problem
sets each require 2 days, then the student delays forBD 5 days.
(b)Arelaxedstudent rolls a fair, 6-sided die in the morning. If he rolls a 1, then he
does his laundry immediately (with zero days of delay). Otherwise, he delays for
one day and repeats the experiment the following morning. LetRbe the number
of days a relaxed student delays laundry. What is ExŒRç?
Example: If the student rolls a 2 the first morning, a 5 the second morning, and a 1
the third morning, then he delays forRD 2 days.
(c)Before doing laundry, anunluckystudent must recover from illness for a num-
ber of days equal to the product of the numbers rolled on two fair, 6-sided dice.
LetUbe the expected number of days an unlucky student delays laundry. What is
ExŒUç?
Example: If the rolls are 5 and 3, then the student delays forUD 15 days.