Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 879 — #887
20.6. Sums of Random Variables 879
Chebyshev’s Theorem gives us the stronger result that
PrŒjT ExŒT j c T 1
c 2 :
The Chernoff Bound gives us an even stronger result, namely, that for any c > 0,
PrŒT ExŒT c ExŒT e .c ln.c/ cC1/ ExŒT :
In this case, the probability of exceeding the mean by c ExŒT decreases as an
exponentially small function of the deviation.
By considering the random variable n T , we can also use the Chernoff Bound
to prove that the probability that T is much lower than ExŒT is also exponentially
small.
20.6.8 Murphy’s Law
If the expectation of a random variable is much less than 1, then Markov’s Theorem
implies that there is only a small probability that the variable has a value of 1 or
more. On the other hand, a result that we call Murphy’s Law 8 says that if a random
variable is an independent sum of 0–1-valued variables and has a large expectation,
then there is a huge probability of getting a value of at least 1.
Theorem 20.6.4 (Murphy’s Law). Let A 1 , A 2 , . . . , A n be mutually independent
events. Let T i be the indicator random variable for A i and define
T WWD T 1 C T 2 C C T n
to be the number of events that occur. Then
PrŒT D 0 e ExŒT :
8 This is in reference and deference to the famous saying that “If something can go wrong, it
probably will.”