Mathematics for Computer Science

19.2. Chebyshev’s Theorem 793

Definition 19.2.2.Thevariance, VarŒRç, of a random variable,R, is:

VarŒRçWWDEx



.RExŒRç/^2



:

Variance is also known asmean square deviation.

The restatement of (19.4) forzD 2 is known asChebyshev’s Theorem^1

Theorem 19.2.3(Chebyshev).LetRbe a random variable andx 2 RC. Then

PrŒjRExŒRçjxç

VarŒRç

x^2

:

The expression ExŒ.RExŒRç/^2 çfor variance is a bit cryptic; the best approach
is to work through it from the inside out. The innermost expression,RExŒRç, is
precisely the deviation ofRabove its mean. Squaring this, we obtain,.RExŒRç/^2.
This is a random variable that is near 0 whenRis close to the mean and is a large
positive number whenRdeviates far above or below the mean. So ifRis always
close to the mean, then the variance will be small. IfRis often far from the mean,
then the variance will be large.

19.2.1 Variance in Two Gambling Games

The relevance of variance is apparent when we compare the following two gam-
bling games.
Game A:We win $2 with probability2=3and lose $1 with probability1=3.
Game B:We win $1002 with probability2=3and lose $2001 with probability
1=3.
Which game is better financially? We have the same probability, 2/3, of winning
each game, but that does not tell the whole story. What about the expected return for
each game? Let random variablesAandBbe the payoffs for the two games. For
example,Ais 2 with probability 2/3 and -1 with probability 1/3. We can compute
the expected payoff for each game as follows:

ExŒAçD 2 

2

3

C.1/

1

3

D1;

ExŒBçD 1002 

2

3

C.2001/

1

3

D1:

The expected payoff is the same for both games, but the games are very different.
This difference is not apparent in their expected value, but is captured by variance.

(^1) There are Chebyshev Theorems in several other disciplines, but Theorem 19.2.3 is the only one
we’ll refer to.