The Law of Large
Numbers and Sample Average
The Law of Large Numbers deals with estimating expected
value from a large number of observations of values of events from the same
population. The Law of Large Numbers will be very valuable in the process of
rolling wave planning, which we will address in the chapter on scheduling.
To proceed we need to define what is meant by the "sample
average":
α(x) = (1/n) * (X1 + X2 + X3 + X4 + X5 + X6 + ...)
where α(x) is the "arithmetic average of a sample
of observations of random variable X,"
using the lower case alpha from the Greek alphabet. α(x) is a
random variable with a distribution of possible values; α(x) will
probably be a different value for each set of Xis that are observed.
We call α(x) a "sample average" because we cannot be
sure that the Xi that we
observe is the complete population; some outcomes may not be in the sample.
Perhaps there are other values that we do not have the opportunity to observe.
Thus, the Xi in the α(x) is but a sample of the total population. α(x) is not the expected value of X
since the probability weighting for each Xi is not in the calculation; that is, α(x) is an
arithmetic average, and a random variable, whereas E(X) is a probability weighted average and a deterministic
nonrandom variable.
Now here is where the Law of Large Numbers comes in. It can be
proved, using a couple of functions (specifically Chebyshev's Inequality and
Markov's Inequality [21]), that as
the number of observations in the sample gets very large, then:
α(x) ≈ E(X) = μ
This means that the sample average is approximately equal to
the expected value or mean of the distribution of the population of X. Since Xi are observations from the same distribution
for the population X, E(Xi) = E(X). That is, all values of X share the same population parameters or expected value
and standard deviation.
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