Approximating the Bell Curve by Pascal's Triangle
1. Fun fact about Pascal’s Triangle
a.  , so dividing all entries by 2n
would make the sum 1.
2. Fun facts about the bell curve.
a. Pictures in books almost always use a different
scale for the x and y axes; if they didn’t, normal
distribution curves would look like a flat line with a tiny bump in
the middle. If you enter the formula into a Texas Instruments
graphing calculator, you can use the ZoomFit command to make the
tiny bump look like the bell curve you see in books. If you
want to see a more realistic (though still stretched) approximation
of bell curve, go to the Bar Graph
applet; the number of attempts can range from 1 to 25, and you can see
both how the top height changes as the number of attempts changes, and
by changing the probability of success, you can see approximations to
normal curves that are not symmetric.
b. The formula for the normal distribution curve is , where  (mu) is the midpoint and
 (sigma)>0 is the standard deviation. f
’(x) = 0 at and f ’’(x) = 0
at ± , the standard deviation; this
is why standard deviation is important and we use the somewhat
awkward formula instead of the more straightforward average of
absolute values, 1/n(sum of |midpoint–data points|).
c. The integral of any bell curve over the entire x-axis
is 1; the integral from –1SD to +1SD 68.27% and from –2SD to +2SD 95.45%, where SD stands for standard deviation.
3. Fun fact about the bell curve and Pascal’s Triangle.
a. The values of the nth row of
Pascal’s Triangle when divided by 2n and properly
spaced make a pretty good approximation to a bell curve. If we
take an even row, we have the central value  /22n; if we space the values out one
unit apart, we get the left edges of a fake Riemann sum of rectangles
equal to 1; if we want to approximate any given normal curve, we can
multiply by some x so that the middle spike is as tall as the
given curve, then space the other spikes at a distance of 1/x
apart, and our fake Riemann sum remains 1. Note again, the points
are a pretty good approximation of the curve; except for the middle
spike, no point in the sequence is guaranteed to be exactly on the
curve.
4. Given any even row of Pascal’s Triangle, call it row 2n,
if we divide by 22n, how closely does some middle
sum of the Riemann rectangles get to 68.27% and 95.45%?
a. Here we use the sequence analog of the second
derivative going to zero, which we have discussed as the difference
triangle; as we go from left to right on the normal curve, when f ’’(x)
= 0, the slopes of the tangents of the curve stop getting steeper and
start becoming less steep. In a sequence, this is analogous to
when sn–sn-1>=sn+1–sn.
b. It turns out on any (n2-2)th
row of Pascal’s Triangle, where n > 2, there will be three
entries in a row, call them sn-1, sn and
sn+1 such that sn–sn-1 = sn+1–sn;
obviously, since Pascal’s Triangle is symmetric on every row, there
are three such entries left of the midpoint and three such entries
right of the midpoint; the formula for finding sn
on the even numbered rows is  for all n > 2.
This means for these rows, we get a particularly good approximation
to the 1SD and 2SD integrals, and the approximations
improve as n gets larger.
c. Likewise, the approximations at the even square
rows are pretty good, but are slightly less than 68.27% and 95.45%,
while the sums on the (4n2-2)th rows are always
slightly higher than 68.27% and 95.45%. Use the slider bar to
check out how the values of the green and non-red areas change, as
well as the actual height of the curve at its tallest point; the
program shows how close the approximations are when the row is either n2
or n2-2.
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