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The sketch below provides an explanation of the observed density of points for the second step of the random walk (number 2 on the previous page). We first note that all orientations of the first step are equally likely. Thus it suffices to consider only one direction. Let that be point 1.
Consider a ring centered at point start of width r. We wish to know the proportion of second steps that will fall within this ring. This proportion will be simply the sum of the two arc angles (a1 and a2) divided by 360°.
But the observed density of points depends on the area of the ring as well as the proportion of points that fall within it. By measure the area of the two circles and taking a difference, we get the area of the ring. The density of points is the proportion divided by the area. That expression is:
You can use Sketchpad to graph this value as a function of distance from the starting point. (Measure the radius of the inner circle. Plot this distance and the computed density as (x, y). Construct the locus of the plotted point as the point that determines the radius of the inner circle moves on the radius segment.)
Does your plot correspond to your observation of the density? Can you give a qualitative explanation of the density?
© 1995 Key Curriculum Press