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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 n^th 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 s_n–s_n-1>=s_n+1–s_n.

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 s_n-1, s_n and s_n+1 such that s_n–s_n-1 = s_n+1–s_n; 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 s_n 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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