Concrete mathematics : a foundation for computer science

404 DISCRETE PROBABILITY
Therefore the pgf for P itself is
where
GP(z) = ‘&~GP,I;(z’I
k=l
= 2s ~-
n-l+2
( m > ’
(8.92)
S(z) = Sl + s2z + s& + . . . + S,P’ (8.93)
is the pgf for the search probabilities sk (divided by z for convenience).
Good. We have a probability generating function for P; we can now find
the mean and variance by differentiation. It’s somewhat easier to remove the
z factor first, as we’ve done before, thus finding the mean and variance of
P - 1 instead:
Therefore
F(z) = Gp(z),‘z = S(m-m+f) ;
F’(z) = ;S’(+) ;
F”(z) = -&“(!+) .
EP = 1 + Mean(F) = 1 + F’( 1) = 1 + m-’ Mean(S) ; (8.94)
VP = Var(F) = F"(l)+F'(l)-F'(l)'
= rn-‘S”(1) +m-‘S’(1) -m~2S’(1)2
= rnp2 Va.r(S) + (rn-’ - m-*) Mean(S). (8.95)
These are general formula,s expressing the mean and variance of the number
of probes P in terms ‘of the mean and variance of the assumed search
distribution S.
For example, suppose we have sk = l/n for 1 6 k 6 n. This means
we are doing a purely “ran.dom” successful search, with all keys in the table
equally likely. Then S(z) is the uniform probability distribution U,(z) in