P^2+Σ2P: 2023

2023年6月13日火曜日

Hoeffding’s inequality

8/10 is 8 heads in each 10 times. This is qite rare.

80% of the heads are in 99.9%. The red is the area.

t≧0

In this case, t is 0.8, and -t is 0.2. They are symmetry. 1/100 is the possibilty to have 80% of the head. This is very difficlt like finding rich men.

Var(Z)＜∞

∴

You may know Bernoulli trial.

n=10, and k=8.

P(X=8)=0.044≒4/100.

Therefore, you may have 1～4 of (8/10) heads in each 100 times.You can't predict anything because of Zero.

2023年6月6日火曜日

Central limit theorem

Whenever you toss your coins, you are close to normal distribution.

You can see Gaussian function.

μ is the average, and σ^2 is the distribution.

Y is E(X).

I tossed coins 100 times.

NORMDIST(x,0.5,(x-0.5)^2,0)

You may have 80% of the head once (1.7%). When you are around Y which is the average, it is almost Zero. You need to go far beyond that. It is quite rare.0.5 is null, and 0.8 and 0.2 are symmetry.

NORMDIST(0.8,0.5,(0.8-0.5)^2,1)≒0.999

X=0.6 is 100%

The average is normal and Zero, but 9/10 and 10/10 is the difficulty. It is narrower and close to Zero. You may talk about singularity.

2023年6月5日月曜日

Markov’s Inequality

It is hard to see rich men.

This is called Markov’s Inequality. E[X] is the average, although I don't reach it. X is a discrete random variable.

∴

a＞0

2023年6月2日金曜日

Gaussian function

This is an even function which is symmetry by the Y-axis.

∴

This is the inflection point. You still see the square.

f(0)=1

The average is the top which is called normal distribution.

μ is the average, and σ^2＞0. R=(-∞,∞)

2023年2月6日月曜日

Euclidean distance

This is taxicab geometry. You move from X to Y. Each square is 1.

Red:1+2+2+1+2+2=10

Green:5+5=10

They are the same line.

This is called Manhattan distance. n is the dimension.

Then, this is Euclidean norm.

This is 1*1 squares.

∴

There are 25 suares, so K=25. √50≒7.07.

50=5^2+5^2. This is the green.

2023年2月5日日曜日

Approximate Carathéodory’s Theorem

Higher dimensions are chaotic. However, you close to Zero, although you keep expanding.

This is almost 1-1=0. k go to ∞, which is called Euclidean norm.

Then you pile squares.

They are almost circles, and cubes are also balls. They are closed.

However, higher dimensions are opened, so they are chaotic.

2023年1月26日木曜日

The empirical method of Maurey

It is hard to define boundary mathematically. Zeta function is well known. ζ(s)=0

You need to fill all spaces for integration.

A⊂R^d

x is the convex hull of A.

When λn, n>d+1. Then, n-1 go to 0.

x2-x1=0.00000000000000000009. μ is the vector.

x1	x2	・・・・	xn
λ1(0.9)	λ11(0.00000000009)
λ2(0.09)	λ12(0.000000000009)
λ3(0.009)	λ13(0.0000000000009)
λ4(0.0009)	λ14(0.00000000000009)
λ5(0.00009)	λ15(0.000000000000009)
λ6(0.000009)	λ16(0.0000000000000009)
λ7(0.0000009)	λ17(0.00000000000000009)
λ8(0.00000009)	λ18(0.000000000000000009)
λ9(0.000000009)	λ19(0.0000000000000000009)
λ10(0.0000000009)	λ20(0.00000000000000000009)

λ10 is λ1 because of x1 which is the same group. You use Σ in your Excel. μ10 is also μ1. This is 0.0000000001. Then you keep expanding, but you close to Zero. However, you never reach Zero.