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

Theorem: If X i , 1;i;n, are random variables on S and if a,b,R, then
1. E(X 1 +X 2 +…+X n ) = E(X 1 )+E(X 2 )+…+E(X n )
2. E(aX i +b) = aE(X i ) + b
Proof: Use mathematical induction (base case is n=2) for 1 and using the
definitions for 2.
Note: The linearity of E is extremely convenient and useful.
Theorem: The expected number of successes when n Bournoulli trials is
performed when p is the probability of success on each trial is np.
Proof: Apply 1 from the previous theorem.
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Notes:
• The average case complexity of an algorithm can be interpreted as the
expected value of a random variable. Let S={a i }, where each possible input
is an a i . Let X be the random variable such that X(a i ) = b i , the number of
operations for the algorithm with input a i . We assign a probability p(a i )
based on b i . Then the average case complexity is E(X) = " p(a i
)X(a i
).
a i
!S
• Estimating the average complexity of an algorithm tends to be quite
difficult to do directly. Even if the best and worst cases can be estimated
easily, there is no guarantee that the average case can be estimated without a
great deal of work. Frankly, the average case is sometimes too difficult to
estimate. Using the expected value of a random variable sometimes
simplifies the process enough to make it doable.
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