Class summary:
Wednesday, 4/2
I began Section 3.3, which deals with two topics: variance and
standard deviation, and normal approximation. I went over the
definitions, interpretations, and properties,
of variance and std. deviation, and
did some simple computations and easy proofs involving these concepts.
During the final part of the hour, I began the second topic, normal
approximation, which is of fundamental importance in probability and
statistics and on which we'll spend at least another two hours.
I passed out a handout on normal approximation, and went over
the formulas in this handout. Normal approximation has come up earlier in
a very specific context - namely, as an approximation to a particular
distribution - the binomial distribution. However, normal
approximation arises in a much more general context - namely, whenever
a sum of many independent, identically distributed random variables is
involved. This section deals with normal approximation in this general
context.
The back page of the class handout (copied from pp. 200-201 of the
Pitman text) has two examples showing clearly
how the normal distribution takes
shape when one starts with an arbitrary distribution and
takes a sum of independent random variables having this distribution.
As the number of terms in the sum increases, the distribution of the
sum resembles more and more that of the normal distribution.
You can confirm this behavior yourself using
this Java application, which lets you specify the initial
distribution and then displays the distribution of the corresponding
sums. Regardless of the shape of the initial distribution, the sums
sooner or later take on the shape of the normal curve.
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