goal
what should we learn
communication system
Source Encoder
Converts the raw source (text, audio, image) into a sequence of bits .
Channel Encoder
Adds redundancy to protect against noise.
Modulator
Converts the encoded bits into waveforms/signals that can physically travel through the channel.
Channel + Noise
The medium distorts the signal (attenuation, interference, random noise).
Demodulator
Recovers the bit sequence from the noisy received signal.
Channel Decoder
Uses r...
Joint hypothesis testing
Simple null hypothesis → involves a restriction on one sign (<,=,>) only
Joint null hypothesis → involves two or more restrictions at the same time
Testing the effect of advertising: The F-test
restricted model :
An unrestricted model is the “full” regression specification, where you estimate all parameters freely without imposing any restrictions. For example, if your regression is
then the unrestricted model estimates all at once.
A restricte...
Self -informationIn information theory and statistics, “surprise” is quantified by self-information. For an event x with probability p(x), the amount of surprise (also called information content) is defined as
This definition has several nice properties:
Rarity gives more surprise: If p(x) is small, then I(x) is large — rare events are more surprising.
Certainty gives no surprise: If p(x)=1, then I(x)=0. Something guaranteed to happen is not surprising at all.
Additivity for independence...
Definition
Mutiplication - Recursion
Mutiplication - iteration
Observation
类比思路
TOWERS OF HANOI1234567891011121314151617def tower(n,fro,to,spare): if n==1: print("from %c to %c" % (fro,to)) else: tower(n-1,fro,spare,to) tower(1,fro,to,spare) tower(n-1,spare,to,fro)# 测试汉诺塔函数tower(3, 'A', 'B', 'C')
fro:起点to：终点spare：空闲的柱子
123456789101112131415161718192021222324252627def permuation(num): res=[] def backtrack(start): if start==len(num): re...
O-notation注意：计算机教材的写法 是一个元素， 代表属于，是一个集合
定义回顾
例子 1：
这里
要检查是否成立：
化简：
当，，所以总能找到 c。
比如选 c=1，那么只要 ，。
所以
例子 2：
这里
检查是否成立：
化简：
这要求：
但随着 n 越来越大，不可能存在一个有限常数 c 让不等式一直成立。
所以 。
Macro substitution第一条：左边元素，右边集合，左边元素一定存在于右边。第二条：两边都是集合，左边整体是右边的子集，左边任意元素都能在右边被找到。
Operations of 𝑶-notation
，
-notation
-notation and -notation
Transitivity
卖出以及何时卖出费雪认为，卖出股票的三个理由：①、当初买进行为犯下错误，某特定公司的实际状况显著不如原先设想那么美好。在某种程度上，要看投资者能否坦诚面对自己。②、当成长股成长潜力消耗殆尽，股票与持有原则严重脱节时，就应该卖出。③、有更加前景远大的成长股可以选择。
费雪认为投资者不应该因为空头市场的担忧而卖出股票（恐惧熊市的到来）。他认为这样做无异于要求投资者知道空头市场何时出现，以及何时结束（在空头市场底部买回股票）。但通常情况是投资者卖出后，空头市场并未出现，市场继续上扬。等到空头市场真的来临时，却从来没有见过比卖出价格更低的位置买回相同股票的投资者。通常的情况是，股价并没有跌回卖出价，但投资者仍苦苦等待，或者股价真的一路下挫跌过卖出价，他们却又忧虑别的事情而不敢买回。
费雪提醒：投资者不应该因为手上的股票涨幅过大就卖出股票。因为，“涨幅过大”、“估值过高”，都是非常模糊的概念。没有证据表明多高的估值或者涨幅才是最高。
投资者的“五不原则”原则一：不买处于创业阶段的公司。（注：中国式创业板其实不属于创业阶段的公司）
费雪认为创业阶段的公司，投资者只能看到它的运作蓝图，并猜测它...
多元回归
以 multiple linear regression with two predictors 为例
$$\tilde{y}i = y_i - \bar{y},\quad \tilde{x}{i2} = x_{i2} - \bar{x}2,\quad \tilde{x}{i3} = x_{i3} - \bar{x}_3 .$$
important point: In this model, as in many others, it is important to recognize that the model is an approximation to reality in the region for which we have data; including an intercept improves this approximation even when it is not directly interpretable
• Estimated regression models describe the relationship between the ...