From Algorithms to Z-Scores - matloff - University of California, Davis

58 CHAPTER 3. DISCRETE RANDOM VARIABLES
This is indeed true, which we will now derive. First we’ll need some facts (which you should file
mentally for future use as well):
Properties of Geometric Series:
(a) For any t = 1 and any nonnegative integers r ≤ s,
s
i=r
t i r 1 − ts−r+1
= t
1 − t
(3.81)
This is easy to derive for the case r = 0, using mathematical induction. For the general case,
just factor out t s−r .
(b) For |t| < 1,
∞
i=0
t i = 1
1 − t
To prove this, just take r = 0 and let s → ∞ in (3.81).
(c) For |t| < 1,
This is derived by applying d
dt
Deriving (3.80) is then easy, using (3.83):
∞
it i−1 =
i=1
to (3.82).5
EW =
= p
1
(1 − t) 2
(3.82)
(3.83)
∞
i(1 − p) i−1 p (3.84)
i=1
= p ·
= 1
p
∞
i(1 − p) i−1
i=1
1
[1 − (1 − p)] 2
5 To be more careful, we should differentiate (3.81) and take limits.
(3.85)
(3.86)
(3.87)