The Monty Hall Problem

https://www.youtube.com/watch?feature=player_embedded&v=mhlc7peGlGg#!

機率問題

Q: 有一試劑對某疾病有九成五的正確性，此疾病每千人有一人感染。
     你測得的結果為陽性，你得病的機率為多少？

A: 2%（若測一千人，有五十人會是陽性，而只有一人得此病。）

為什麼測一千人會有五十人是陽性？

要用的數學

• Linear Algebra

線性代數，可以用矩陣表示運算過程。

高斯消去法              check
高斯約旦消去法      check

• The Calculation Difference Between Real Number And Matrix

1. AB is not equal to BA

2. AC=BC,
    if C is not equal to 0,
    A is not equal to B.

(AB)T=BTAT
A*AT=symmetric
(AB)-1=B-1A-1

Ito's Lemma

Stochastic Method

Brownian Emotion

The Pricing of Options and Corporate Liabilities

Fischer Black/ University of Chicago
Myron Scholes/ MIT

The Valuation Assumption
a) The short-term interest rate is known and is constant through time.
b) The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus the distribution of possible stock prices at the end of any finite interval is lognormal. The variance rate of the return on the stock is constant.
c) The stock pays no dividends or other distribution.
d) The option is "European", that is, it can only be exercised at maturity.
e) There are no transaction cost in buying or selling the stock or option.
f) It is possible to borrow any fraction of the price of a security to buy it or to hold it, at the short-term interest rate.
g) There are no penalties to short selling.

幹後面的數學我看不懂...........

Sampling(statistics)

Did you remember the question we got when we did the body measurement last time?

The question is like "Is this sampling random? Concerning about the gathering of the elemental selection."

I found out there are four types of random sampling:

1. simple random sampling: all elements in the frame have an equal probability of selection. 

    ->not this one

2. systematic sampling: selecting the elements according to a planed scheme, like n+5, n+10, n+15......

    ->not this one either

3. stratified sampling: if the population has a number of distinct categories, then the population can be separated by these categories into different "strata." then these stratum are considered to be some independent sub-population, out of which individual elements can be randomly selected. then, we just  do the simple random sampling in each strata. 

    -> ahha, I think is this one!

4. cluster sampling: cluster sampling is sort of the same with stratified sampling; the different part is that the elements in each cluster must be heterogeneous. that way, each cluster can be taken as a whole population, and all we have to do is a simple random sampling in only one cluster.

    -> not this one

therefore, in this case, they can do the stratified sampling separating Taiwan into some geographical regions, like north, middle, and south. and do simple random sampling in each.（幹太累了我不想打英文了）不過我覺得還是會不準，因為他只有做北中南，缺了東部。然後還有一個主要的問題，集中做都市的樣本不是random selection阿，除非他把北中南三區內再區分各clusters才可以吧.................