同學(xué)們大家好，學(xué)長(zhǎng)今天又為大家?guī)?lái)了關(guān)于羅徹斯特大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)專(zhuān)業(yè)的相關(guān)知識(shí)點(diǎn)，對(duì)于學(xué)姐來(lái)說(shuō)，數(shù)學(xué)在眾多領(lǐng)域?qū)W科中起著至關(guān)重要的作用，這種作用不可替代，需要同學(xué)們認(rèn)真對(duì)待這門(mén)學(xué)科。

那么數(shù)學(xué)和統(tǒng)計(jì)學(xué)之間既然有聯(lián)系，學(xué)姐今天就位大家以這兩個(gè)知識(shí)點(diǎn)來(lái)詳細(xì)的總結(jié)歸納，究竟應(yīng)該如何才能更好的領(lǐng)悟數(shù)學(xué)知識(shí)，這是非常重要的，也是今后的重點(diǎn)。

無(wú)限集合和基數(shù)

有限集合

有限集合不是空的就是有 n 元素。如果一套有 n 元素，與自然數(shù)集合存在一一對(duì)應(yīng)關(guān)系，{1,2,3,...,n} 在哪里 n∈N.

例如，{p,q,r} 可以與...一一對(duì)應(yīng) {1,2,3}。其中一個(gè)功能是 p→1q→2r→3.

如果設(shè)置 S 有 n 元素 |S|=n。也 |?|=0.

無(wú)限集合

無(wú)限集合是不能與一一對(duì)應(yīng)的非空集合 {1,2,3,...,n} 對(duì)任何人 n∈N。

基數(shù)

For all sets A,B,C,if |A|=|B| and |B|=|C| then |A|=|C|

相同基數(shù)

如果設(shè)置 A 和設(shè)置 B 有相同的基數(shù)，那么就有一個(gè)一一對(duì)應(yīng)從集合 A 設(shè)置 B。

對(duì)于有限集合，集合的基數(shù)是集合中元素的數(shù)量。

可數(shù)無(wú)限的

一套 A 存在可數(shù)無(wú)限的當(dāng)且僅當(dāng)設(shè)置 A 具有與相同的基數(shù) N (自然數(shù))。

如果設(shè)置 A 存在可數(shù)無(wú)限的，那么 |A|=|N|.

此外，我們將可數(shù)無(wú)限集合的基數(shù)指定為 ?0 (“aleph null”)。

|A|=|N|=?0.

可數(shù)的

一套是可數(shù)的當(dāng)且僅當(dāng)它是有限的或者可數(shù)無(wú)限的。

無(wú)限的

不可數(shù)的集合是無(wú)數(shù)的或者無(wú)窮無(wú)盡。

可數(shù)集的任何子集都是可數(shù)的。

如果 S 是無(wú)限的 A?S 然后 A 是可數(shù)的。

超限數(shù)

如前所述，?0 用來(lái)表示可數(shù)集合的基數(shù)。超限數(shù)用來(lái)描述“更高&更高”不定式的基數(shù)。

?0=|N|=|Z|=|Q| 可數(shù)無(wú)限集的基數(shù)。

?1=|R|=|(0,1)|=|p(N)| “最低”不可數(shù)無(wú)限集合的基數(shù);也被稱(chēng)為“連續(xù)體的基數(shù)”。

?2=|p(R)|=|p(p(N))| 下一個(gè)不可數(shù)無(wú)限集合的基數(shù)

由此我們可以看出 2?0=?1。

其他奇怪的數(shù)學(xué)可以用超限數(shù)來(lái)完成，例如 ?1+?0=?1.

集合不能映射到它的冪集合的證明類(lèi)似于羅素悖論，以伯特蘭·羅素命名。

這連續(xù)統(tǒng)假設(shè)是這樣一種說(shuō)法，即沒(méi)有基數(shù)嚴(yán)格介于 N and R。連續(xù)統(tǒng)假說(shuō)實(shí)際上是從連續(xù)統(tǒng)開(kāi)始的推測(cè)，直到它被證明與實(shí)數(shù)系的通常公理一致(由庫(kù)爾特·哥德?tīng)栐?940年提出)，并且獨(dú)立于那些公理(由保羅·寇恩在1963年提出)。

統(tǒng)計(jì)分支

描述性統(tǒng)計(jì)

描述統(tǒng)計(jì)學(xué)是統(tǒng)計(jì)學(xué)的一個(gè)分支，用于通過(guò)圖表或其他統(tǒng)計(jì)方法來(lái)描述數(shù)據(jù)。當(dāng)我們有很多數(shù)據(jù)并想適當(dāng)?shù)乜偨Y(jié)它時(shí)，這就是我們做的統(tǒng)計(jì)類(lèi)型。

推理統(tǒng)計(jì)

匯總數(shù)據(jù)

有多種方法可以總結(jié)數(shù)據(jù)集:

分配表

原始數(shù)據(jù)圖表

樣本統(tǒng)計(jì)，如平均值、中值、模式、標(biāo)準(zhǔn)誤差和標(biāo)準(zhǔn)偏差

基于平均值的圖表，帶有誤差線(xiàn)，表示標(biāo)準(zhǔn)誤差或標(biāo)準(zhǔn)偏差

簡(jiǎn)單描述性統(tǒng)計(jì)

描述性統(tǒng)計(jì)是描述一組數(shù)據(jù)的數(shù)字和過(guò)程。最常見(jiàn)的描述性統(tǒng)計(jì)側(cè)重于確定數(shù)據(jù)的“平均值”。然而，有不止一個(gè)“平均值”，所以我們?cè)谡业剿鼈儠r(shí)必須具體。描述“平均值”的值有:

中值-最中間的數(shù)據(jù)，當(dāng)所有數(shù)據(jù)按數(shù)量排列時(shí)

模式-集合中最常見(jiàn)的數(shù)據(jù)

可以想象，數(shù)據(jù)集很少都是一個(gè)值。因此，我們需要描述數(shù)據(jù)是如何圍繞平均值排列的。描述平均值周?chē)鷶?shù)據(jù)變化的值有:

標(biāo)準(zhǔn)差描述的是，對(duì)于整個(gè)人群，整個(gè)人群數(shù)據(jù)集的離散度。

標(biāo)準(zhǔn)誤差描述了代表性樣本數(shù)據(jù)集的標(biāo)準(zhǔn)偏差。

Small pizza is 8" and large pizza is 16". Are we getting the same amount of pizza if we order two small pizzas instead of one large one?

We all have faced a situation when need to be able to change from one unit of measurement to another unit of measurement. In this section,we discuss a method of converting units called dimensional analysis.

Time

1 day = 24 hours,

1 hour = 60 minutes,and

1 minute = 60 seconds.

Length

Imperial measurement

1 inch

1 foot= 12 inches

1 yard= 3 feet=36 inches

1 mile =5289 feet

Metric

millimeter (mm)

centimeter (cm),1 cm = 10 mm

meter (m),1m= 100 cm= 1000 mm

kilometer (km). 1 km= 1000 m

Basics of Statistics

Like most people,you probably feel that it is important to "take control of your life." But what does this mean? partly it means being able to properly evaluate the data and claims that bombard you every day. If you cannot distinguish good from faulty reasoning,then you are vulnerable to manipulation and to decisions that are not in your best interest. Statistics provides tools that you need in order to react intelligently to information you hear or read. In this sense,statistics is one of the most important things that you can study.

Here are some claims that you may have heard on several occasions. (We are not saying that each one of these claims is true!)

4 out of 5 dentists recommend Dentyne.

Almost 85% of lung cancers in men and 45% in women are tobacco-related.

Condoms are effective 94% of the time.

Native Americans are significantly more likely to be hit crossing the streets than are people of other ethnicities.

people tend to be more persuasive when they look others directly in the eye and speak loudly and quickly.

Women make 75 cents to every dollar a man makes when they work the same job.

A surprising new study shows that eating egg whites can increase one's life span.

people predict that it is very unlikely there will ever be another baseball player with a batting average over 400.

There is an 70% chance that in a room full of 30 people that at least two people will share the same birthday.

79.48% of all statistics are made up on the spot.

Before going further,however,two distinct meanings of the word “statistics” should be clarified. First,“statistics” could mean the academic discipline,a field of study,involving the collection,presentation,interpretation,and use of data. When used in this sense,the word is considered singular (though it looks plural). Second,in a narrower sense,a number collected in a survey or study is referred to as a “statistic.” For instance,the average height of a sample of 100 people would be a (one) statistic. This is a singular word (of course,two or more of these numbers would be referred to as “statistics”). Be aware of the difference when you encounter this word.

上面這些數(shù)學(xué)和統(tǒng)計(jì)知識(shí)都是學(xué)姐千辛萬(wàn)苦在找來(lái)的呦，大家可能覺(jué)得學(xué)姐的這些知識(shí)點(diǎn)太簡(jiǎn)單了，還想要更多，希望可以給出更加專(zhuān)業(yè)，更加有針對(duì)性的內(nèi)容，可是學(xué)姐真的能力有限，美國(guó)留學(xué)生輔導(dǎo)相關(guān)的深入專(zhuān)業(yè)知識(shí)，找留求藝的老師來(lái)幫忙就可以啦，感謝大家對(duì)學(xué)姐的支持，祝同學(xué)們?cè)缛胀瓿勺约旱膶W(xué)業(yè)。