Borwein Integrale: Misleading Patterns – The Science Spectrum

They follow fixed rules: in the first step, people can jump to any number between -1 and 1. The interval corresponds to the first prime factor 12 1-1 = 1 in the sine of the integral. In the second step, the range of the statisticians is only ½, which means the second prime factor (12 2-1) represents and so on. after n They can only take one more step 12n-1 reach a far point.

As it turns out, the percentage of all crazy statisticians who post a file n– The third step on the zero point, the value nBorwein . integrals men (except for one constant). So to understand the misleading pattern that has surprised many in the past, one must understand how statisticians are distributed over the number line.

At first, all statisticians are at zero, then they are left free. Since they can reach any point between -1 and 1 with the same probability in the first step, they are spread out evenly in this period. Therefore, there are 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 infinitely many statisticians at each point, so the value in the evenly distributed interval corresponds to the largest possible value.

Random walker after the first step

In the second step, they continue to migrate evenly distributed, but this time they can move up to ⅓ Remove from their current location. This changes the distribution along the number line. Those who were on the edge of the distribution after the first move, say near 1, can now beat that limit. Since the space for numbers greater than one was previously empty, a directional flow occurs: some marginal statisticians are now entering the once uninhabited area. On the contrary, no statisticians returned from the empty area (because there was no one).

This reduces their share in periods [23, 43] And the [–43, –23], so the distribution is flattened at the edges. between the points –23 And the 23 On the other hand, pedestrians are still evenly distributed, with many leaving the area like new people. So the value in the range still corresponds to the maximum that can be achieved. Since the second Borwein integral depends only on the proportion of statisticians that are at zero point, the result remains unchanged – even if the distribution takes a different shape overall.

Random walking unit after the second step

In the third step, statisticians can choose the maximum 15 Move to the right or left. So there is only between the two points – (1 – 1315) and 1 – 1315 even distribution. That is, the distribution of statisticians on the number line is getting wider. After each step, the evenly distributed area becomes smaller until it finally disappears completely.

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