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)＜∞

∴

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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.

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).

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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.

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

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=(-∞,∞)