82 1. Streaks
algebra (best performed by a computer algebra package) is as follows.
If p = 1/2 and k is odd, then
rn,k =
1
2n−1
(1 4pq)
(k−1)/2
v=0
−1/2
v
(1
4pq)−1/2
4pq
1 4pq
v

n−1
u=1
u odd
n 1
u
u/2
(k 1 2v)/2
(1
4pq)u/2

4pq
1 4pq
(k−1−2v)/2
+
n−1
u=0
u even
n 1
u
u/2
(k 1)/2
∗(1
4pq)u/2
4pq
1 4pq
(k−1)/2
,
while if p = 1/2 and k is even, then
rn,k =
1
2n−1
(4pq)
(k−2)/2
v=0
−1/2
v
(1
4pq)−1/2
4pq
1 4pq
v

n−1
u=1
u odd
n 1
u
u/2
(k 2 2v)/2
(1
4pq)u/2

4pq
1 4pq
(k−2−2v)/2
.
If p = 1/2, the expression for rn,k is much simpler (see Exercise 3).
These expressions were used to generate Figure 1.
We can use the expression for r(x, y, p) to calculate the mean (and
variance) of the distribution. We recall that for fixed n, the mean of
the distribution {rn,k} equals
n
k=1
krn,k .
The value of this sum can be obtained from the generating function
r(x, y, p) by using calculus. If we compute the partial of r(x, y, p)
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