Piico의 일상

Today we talk about what Pairwise independence, mutual independence, k-wise independence is, and discuss how we can artificially create $2^b-1$ pairwise independent random variables using only $b$ coin flips.  Definitions: MI, PI, k-wise independenceIndependence between every random variable is known as mutual independence. Recall that MI is:\[    \mathbb{P}[\bigcap_{i \in I} E_i] = \prod_{i \in..

Expectation Argument LemmaLast post, we've explored a basic argument using probabilities of the event that a certain object with desirable properties exist. Another way to do the same is the Expectation Argument. This argument uses the expected value of a random variable $X$ in probability space $S$.The following lemma allows us to use such an argument:Lemma 1: Given a probability space $S$ and ..

The motivation of the probabilistic method is to show the existence of an object. The basic idea is to show that there is greater than zero probability for that object in a probability space $S$. There are various methods such as the argument with expectation and derandomization. We will talk about them in future sections. In this post we explore how to make a basic argument about the existence ..

Laplace가 probability를 이렇게 정의하였다:"The probability that A occurs" $:= \mathbb{P}[A] = \frac{|A|}{|S|} = \frac{|A|}{n}$where $S$ is the sample space, $A$ is a subset of $S$, and $|S| = n$. 요즘은 중학교에서도 배우는 내용이겠지만 처음 정의됐을 때 만큼은 혁신적이었을 것이다. Probability theory는 도박류 게임을 연구하기 위해 1700~1800년대에는 많은 관심을 끌었다. 지금은 공학에서 필수적으로 배우는 학문 중 하나이다. 정의는 쉽게 이해가 되지만, Counting문제와 비슷하게 많이 까다로울 때가 많다. From a standard deck of ..