Geometrical analogies to statistical distribution functions?

From: ifignow (ifignow_at_hotmail.com)
Date: 06/28/04 

Date: Mon, 28 Jun 2004 22:23:44 GMT

I just realized this in bed, but the normal distribution corresponds to the
cross-sectional height of a parallelogram.

e.g. The histogram of the sum of two dice approximates the normal
distribution. The more sides the two dice have, the closer the histogram
will look to a normal distribution. If the dice have infinitely many sides
(a continuous distribution), the histogram will be a normal distribution.

Discrete case: you can plot the outcome of two dice on a *** of graph
paper. e.g. two six-sided dice:

Let the x-axis be the value of the first die, the y-axis be the sum of the
two dice. At x=1, we would have a bar that starts at (1,2) extending to
(1,7). At x=2, we would have a bar that starts at (2,3) extending to (2,8).
... At x=6, we would have a bar that starts at (6,7) extending to (6,12).
In the end, we will have bars stacked roughly in the shape of a
parallelogram, and the histogram will correspond to the cross-sectional
width of the parallelogram at a given value of y, from y=2 to y=12. As we
increase the number of sides on the dice to infinity, the stack of bars will
converge to a parallelogram, whose height as a function of x will look like
the normal distribution.

I have not seen such a geometrical analogy mentioned in any stat textbook
I've read. Are there other geometrical analogies to other kinds of
distributions?

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