Buffon injection experiment · intuitive understanding of mathematics · basic knowledge is very important

Starting with a matrix67 blog's mathematical question

Http://www.matrix67.com/blog/archives/2494

Quotations from M Daniel --
   "The really sad thing about mathematics learning is that I remember a magical and great theorem and understood its most rigorous derivation process, but I never understood it intuitively. Although strict derivation is necessary and intuitive understanding is often inaccurate, it is not a cool thing to understand if we can understand an explanation that makes the theorem very clear for a moment, it is also helpful to have a more thorough understanding of the theorem and use it more skillfully."

   For example, the relationship between the area of the circle and the perimeter is M. (Relationship between Square area and perimeter ).

   I remember a lot of friction when I talked with a friend about the probability of weapon upgrades in online games. I studied a complex (but correct) Formula One day, but the result made him very disdainful, because he thought there must be an intuitive and SIMPLE algorithm. At that time, I thought he thought this was because he had a poor math foundation =. It turns out that I was wrong. -- For this question, write another article to record it.

   Repeat the Buffon issue.

    

   In the Buffon needle placement experiment, we assume that a group of parallel lines with 1 spacing are painted on the floor. If you throw a needle with a length of 1 to the ground, the probability of the needle joining a parallel line on the floor is 2/PI. The answer is a bit strange. One is pi, and the other is because it is too concise.

 

Note that, after a wire with Pi length is bent into a circle with a diameter of 1, it is thrown to the ground and there are always two intersections between it and this set of parallel lines. That is to say, the C times of PI is equal to 2, that is, C is equal to 2/PI.

After all, pay attention to the red part-one of the most striking properties of expectation is that E (a + B) = E (A) + E (B ), whether a and B are independent. I am very puzzled about this nature, and have not yet fully understood how to understand or prove this nature. First paste an online proof to see:

The mathematical expectation of the sum of two random variables equals the mathematical expectation of the two random variables. (Easy to hide. It is easy to describe it in text .)

Certificate: E (x + y) = sig (I) sig (j ){(AI + BJ) · PIJ} = Sig (I) {ai · PIJ} + sig (j) {BJ · PIJ}

          =
SIG (I) {ai · PI} + sig (j) {BJ · PJ}

          = Ex + ey

 

   I was not good at mathematics. I did not understand the proof process last time. The general understanding of this time is that only simple multiplication and addition methods are expected in the definition. As long as the allocation rate and exchange rate are used to reverse, the desired result will be obtained. Only dual Sigma is unfamiliar.

Paste it first and understand it slowly. Basic knowledge is important.

Intuitively expected, this property is very useful =