Concrete mathematics : a foundation for computer science

378 DISCRETE PROBABILITY
And we can also make an estimate of the variance, using the formula
\ix 1
x: + x: + + ;y’n _ (X, + X2 + ‘. + X,)2
n - l n(n-1)
(8.20)
The (n ~ 1) ‘s in this formula look like typographic errors; it seems they should
be n’s, as in (8.1g), because the true variance VX is defined by expected values
in (8.15). Yet we get a better estimate with n - 1 instead of n here, because
definition (8.20) implies that
E(i/X) = VX.
Here’s why:
E(\;/X) =&E( tx: -
k=l
k=l
1 n
=n
- l(x W2)
k=l
- k f f (E(Xi’lj#kl+ E(X')Lj=kl))
j=l k=l
= &(nE(X’) - k(nE(X’) +n(n- l)E(X)'))
= E(X')-E(X)“ = VX
; f f xjxk)
j=l k=l
(8.21)
(This derivation uses the independence of the observations when it replaces
E(XjXk) by (EX)‘[j fk] + E(X’)[j =k].)
In practice, experimental results about a random variable X are usually
obtained by calculating a sample mean & = iX and a sample standard de-
viation ir =fi, and presenting the answer in the form ‘ fi f b/,/i? ‘. For
example, here are ten rolls of two supposedly fair dice:
The sample mean of the spot sum S is
fi = (7+11+8+5+4+6+10+8+8+7)/10 = 7.4;
the sample variance is
(72+112+82+52+42+62+102+82+82+72-10~2)/9z
2.12