Misleading Patterns

They follow fixed rules: In the first step, people can jump to any number between -1 and 1. The interval corresponds to the first prefactor ¹⁄_{2 1-1} = 1 in the sine function of the integral. In the second step, the range of the statisticians is only ½, which means the second prefactor (¹⁄_{2 2-1}) represents and so on. After n They can only step one more step ¹⁄_{2 n-1} reach distant point.

As it turns out, the proportion of all the insane statisticians posting the n-th step are on the zero point, the value of the n-ten Borwein integrals I_n (except for one constant). So to understand the misleading pattern that has amazed so many in the past, one must understand how statisticians are distributed on the number line.

In the beginning, all statisticians are at the zero point, then they are left free. Since they can reach any point between −1 and 1 with the same probability in the first step, they spread out evenly in this interval. There are therefore as many statisticians at the zero point as there are at the edge of the range at plus or minus one – or any point in between. It is important to note that there are an infinite number of statisticians at each point, so the value in the evenly distributed interval corresponds to the largest possible one.

In the second step, they continue to migrate evenly distributed, but this time they can only move by a maximum of ⅓ remove from their current location. This changes the distribution along the number line. Those who were at the edge of the distribution after the first step, e.g. near 1, can now overcome this limit. Since the area of ​​numbers greater than one was previously empty, a directional flow is taking place: some of the marginal statisticians are now entering the once unpopulated area. Conversely, no statisticians come back from the empty zone (because nobody was there).

This reduces their share in the intervals [²⁄₃, ⁴⁄₃] and [–⁴⁄₃, –²⁄₃], so the distribution flattens out at the edges. Between the points –²⁄₃ and ²⁄₃ the walkers, on the other hand, are still evenly distributed, since as many leave the area as new ones. The value in the range therefore still corresponds to the maximum that can be achieved. Because the second Borwein integral only depends on the proportion of statisticians who are at the zero point, the result remains unchanged – even if the distribution takes on a different shape overall.

In the third step, the statisticians can choose a maximum ¹⁄₅ move to the right or left. Therefore there is only between the points – (1 – ¹⁄₃ – ¹⁄₅) and 1 – ¹⁄₃ – ¹⁄₅ an equal distribution. That is, the distribution of statisticians on the number line is getting wider and wider. After each step, the evenly distributed area gets smaller until it eventually disappears completely.