Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Example of linear search average complexity: See page 44 in the class notes for
the algorithm and worst case complexity bound. We want to find x in a distinct
!
set " a
# i
n
$
%
&i=1
. If x =
!
ai , then there are 2i+1 comparisons. If x6 " a i
!
2n+2 comparisons. There are n+1 input types: " a i
!
where p is the probability that x, " a
# i
n
$
%
&i=1
n
i=1
#
n
$
%
&i=1
. Let q = 14p. So,
E = (p/n)! (2i-1) + (2n+2)q
= (p/n)((n+1) 2 + (2n+2)q
= p(n+2) + (2n+2)q.
There are three cases of interest, namely,
• p = 1, q = 0: E = n + 1
• p = q = 0.5: E = (3n + 4) / 2
• p = 0, q = 1: E = 2n + 2
#
n
$
%
&i=1
, then there are
2x. Clearly, p(ai ) = p/n,
111
Definition: A random variable X has a geometric distribution with parameter p if
p(X=k) = (14p) k-1 p for k = 1, 2, …
Note: Geometric distributions occur in studies about the time required before an
event happens (e.g., time to finding a particular item or a defective item, etc.).
Theorem: If the random variable X has a geometrix distribution with parameter
p, then E(X) = 1/p.
Proof:
E(X) =
!
!
i=1
!
i=1
!
i=1
ip(X=i)
= i(1-p) i-1 p
!
= p i(1-p) i-1
!
= pp -2
= 1/p
112