7. Appendix 83
with respect to y, and then set y = 1, we obtain the expression

n=1
n
k=1
krn,kxn
,
which can be written as

n=1
xn
n
k=1
krn,k .
Thus the mean of the distribution for sequences of length n is just
the coefficient of xn in the above expression. The point is that we do
not need to use the formulas for rn,k to calculate the mean. Rather,
we use the closed-form expression for r(x, y, p), and apply the ideas
above to this expression.
If we perform these calculations, we obtain the expression
x
1 x
+
2pqx3
(1 x)2
+
2pqx2
1 x
.
Using the facts that
1
1 x
= 1 + x +
x2
+ . . .
and
1
(1 x)2
= 1 + 2x +
3x2
+ . . . ,
in a suitable interval containing the origin, we can expand each of the
three summands above as series; they are, respectively,
x +
x2
+
x3
+ . . . ,
2pq(x3
+
2x4
+
3x5
+ . . .) ,
and
2pq(x2
+
x3
+
x4
+ . . .) .
Now we can easily write down the coefficient of xn; it is
1 + 2pq(n 2) + 2pq = 1 + 2pq(n 1) ,
if n 2.
There is an easy way to check this. In fact, the calculation below
is an easier way to find the mean in this case, but the above method
can be used to find other moments (including the variance) and the
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