Concrete mathematics : a foundation for computer science

376 DISCRETE PROBABILITY
hence VS = y = 7.5 when both dice are loaded. Notice that the loaded dice
give S a larger variance, although S actually assumes its average value 7 more
often than it would with fair dice. If our goal is to shoot lots of lucky 7’s, the
variance is not our best indicator of success.
OK, we have learned how to compute variances. But we haven’t really
seen a good reason why the variance is a natural thing to compute. Everybody
does it, but why? The main reason is Chebyshew’s inequality ([24’] and If he proved it in
[50’]), which states that the variance has a significant property: 1867, it’s a classic
‘67 Chebyshev.
Pr((X-EX)‘>a) < VX/ol, for all a > 0. (8.17)
(This is different from the summation inequalities of Chebyshev that we encountered
in Chapter 2.) Very roughly, (8.17) tells us that a random variable X
will rarely be far from its mean EX if its variance VX is small. The proof is
amazingly simple. We have
VX = x (X(w) - EX:? Pr(w)
CLJE~~
3 x (X(w) -EXf Pr(cu)
WEn
(X(w)-EX)‘>a
3 x aPr(w) = oL.Pr((X - EX)’ > a) ;
WEn
(X(W)-EX]~&~
dividing by a finishes the proof.
If we write u for the mean and o for the standard deviation, and if we
replace 01 by c2VX in (8.17), the condition (X - EX)’ 3 c2VX is the same as
(X - FL) 3 (~0)~; hence (8.17) says that
Pr(/X - ~13 co) 6 l/c’. (8.18)
Thus, X will lie within c standard deviations of its mean value except with
probability at most l/c’. A random variable will lie within 20 of FL at least
75% of the time; it will lie between u - 100 and CL + 100 at least 99% of the
time. These are the cases OL := 4VX and OL = 1OOVX of Chebyshev’s inequality.
If we roll a pair of fair dice n times, the total value of the n rolls will
almost always be near 7n, for large n. Here’s why: The variance of n independent
rolls is Fn. A variance of an means a standard deviation of
only