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Raindrops games6/14/2023 ![]() ![]() Say, what is the average value of the minimum of the x- and y- coordinates of the raindrops? Of course, the same reasoning indicates that the average value of the y-coordinate is also 1/2. The answer is obviously 1/2, as the x-coordinates fall uniformly on the interval from 0 to 1. For example, what is the average of the x-coordinate of the raindrops? Because the x- and y-coordinates are independent, we can simply ignore the y-coordinate. The blotchy nature of random events notwithstanding, we can still answer simple questions about the random pattern of raindrops. Suppose we restrict our attention to drops which fall within the "unit square" - that is, with x-coordinates and y-coordinates between 0 and 1. The independence property assures us that our estimates of the number of drops in a patch of pavement based on its area will not be confounded by any structure or pattern in the distribution of the drops. This explains why the pattern of drops is "clumpy" or "blotchy." Indeed, the only way for the pattern to be more even would be for successive drops to "know" where the previous drops had landed so they could avoid landing nearby. The location of each random drop is "independent" of the location of the others. This guarantees, for example, that the drops do not cluster along a particular line on the sidewalk, as shown in the next example: The x- and y-coordinates of drops are "independent" - knowing the x-coordinate of a raindrop does not enhance the ability to predict the y-coordinate. (If the area of our square of pavement is not equal to 1, we can divide by the area of the square to convert the raw area measure into a "probability measure," in which the total measure of the region is 1.) ![]() Thus, we can use the area of a patch of sidewalk as a measure of the probability that a drop will land there. The distribution of drops is "uniform" - any two disjoint regions of equal area within the square are equally likely to contain any chosen number of drops. The random pattern of drops reflects three important features: Here are some examples showing the first 100 drops: As each raindrop produces a dark dot on the pavement, the random distribution of the falling raindrops becomes apparent - quite different from a regular grid. Most everyone has had the experience of looking at a concrete sidewalk just as it starts to rain. Averaging Raindrops - an exercise in geometric probability Scott E.
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