Class summary:
Wednesday, 4/9
I concluded the discussion of normal approximation with two more
examples. The first dealt with sums of numbers obtained on 100 rolls
of a die. The second, more important, example showed how normal
approximation can be applied to averages (rather than sums) of random
variables. Regardless of the distribution of the individual random
variables, by taking the average of a large number of independent
r.v.'s with the same distribution, one obtains a random variable that
tends to have a large "spike" at the expected value,
and which after
rescaling has approximately normal distribution.
A typical situation where this phenomenon can be observed and is of
considerable practical importance, is the average
error in a series of repeated measurements, If one assumes that the
errors in the individual measurements are independent and uniformly
distributed, then repeating the measurement a large number of times
and taking the average of the readings results in a much improved
accuracy.
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