CS224d: Deep Learning for NLP Lecture1 概率复习(1)

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因为平时考试，我的报告分数特别低，主要是因为我的英文写作能力特别差，为了练习英文写作，部分博客(比较简单的内容)将用英文写作，望大家闲来无聊时，看看我的博客，指正错误。

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Abstract

The main content is the review of cumulative distribution functions(CDFs), probability mass functions(PMFs) and probability density functions(PDFs).

Axioms of Probability

• $P(A) \geq 0,$ for all $A \in \mathcal{F},$ $\mathcal{F}$ is a set of events
• $P(\Omega) = 1,$ $\Omega$ represents a sample space
• If $A_1, A_2, ...$ are disjoint events, then $P(\cup_iA_i) = \sum_iP(A_i)$

Conditional Probability and Independence

The conditional probability of any event A given B is defined as,

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$ $P(A|B)$ means the probability measure of A when the event B occurs. We can see the event B as a universal set. Two events are called independent if and only if $P(A\cap B)=P(A)P(B)$ or $P(A|B)=P(A)$. Therefore, independence is equivalent to saying that observing B does not have any effect on the probability of A.

Random Variables

We denote random variables using upper case letters $X(\omega)$ or simply $X$. We denote the value that a random variable may take on using lower case letter $x$. Random variables are divided into discrete random variable and continuous random variable. When $X$ can take on only a finite number of values, it is known as a discrete random variable. The probability of the set associated with a random variable $X$ taking on some specific value $k$ is:

$$P(X=k)=P(\{\omega : X(\omega)=k\})$$ When $X$ can take on a infinite number of possible values, it is called a continuous random variable. The probability that $X$ takes on a value between two real constants $a$ and $b$ is:

$$P(a\leq X\leq b)=P(\{\omega : a \leq X(\omega) \leq b\})$$

Cumulative Distribution Functions

A cumulative distribution function is a function $F_X:\mathbb{R}\rightarrow [0,1]$ which specifies a probability measure as,

$$F_X(x)=P(X\leq x)$$ Properties:
- $0 \leq F_X(x) \leq 1$
- $lim_{x \to-\infty}F_X(x)=0$
- $lim_{x \to\infty}F_X(x)=1$
- $x \leq y \Rightarrow F_X(x) \leq F_X(y)$

Probability Mass Functions

When $X$ is a discrete random variable, a simpler way to represent the probability of a random variable is to directly specify the probability of each value that the random variable can assume. A probability mass function is a function $p_X:\Omega \rightarrow \mathbb{R}$ such that:

$$p_X(x)=P(X=x)$$ The notation $Val(X)$ is the set of possible values that the random variable $X$ may assume.
Properties:
- $0 \leq p_X(x) \leq 1$
- $\sum_{x\in Val(X)}p_X(x)=1$
- $\sum_{x\in A}p_X(x)=P(X\in A)$

Probability Density Functions

The CDF $F_X(x)$ is differentiable everywhere for some continuous random variables. The probability density functions is the derivative of the CDF:

$$f_X(x)=\frac{dF_X(x)}{dx}$$ According to the properties of differentiation, for very small $\triangle x$, $P(x\leq X\leq x+\triangle x)\approx f_X(x)\triangle x$
Properties:
- $f_X(x)\geq 0$
- $\int_{-\infty}^{\infty}f_X(x)=1$
- $\int_{x\in A}f_X(x)dx=P(X\in A)$

Reference:http://cs229.stanford.edu/section/cs229-prob.pdf

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