同学们大家好，学长今天又为大家带来了关于罗彻斯特大学数学与统计学专业的相关知识点，对于学姐来说，数学在众多领域学科中起着至关重要的作用，这种作用不可替代，需要同学们认真对待这门学科。

那么数学和统计学之间既然有联系，学姐今天就位大家以这两个知识点来详细的总结归纳，究竟应该如何才能更好的领悟数学知识，这是非常重要的，也是今后的重点。

无限集合和基数

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，与自然数集合存在一一对应关系，{1,2,3,...,n} 在哪里 n∈N.

例如，{p,q,r} 可以与...一一对应 {1,2,3}。其中一个功能是 p→1q→2r→3.

如果设置 S 有 n 元素 |S|=n。也 |∅|=0.

无限集合

无限集合是不能与一一对应的非空集合 {1,2,3,...,n} 对任何人 n∈N。

基数

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

相同基数

如果设置 A 和设置 B 有相同的基数，那么就有一个一一对应从集合 A 设置 B。

对于有限集合，集合的基数是集合中元素的数量。

可数无限的

一套 A 存在可数无限的当且仅当设置 A 具有与相同的基数 N (自然数)。

如果设置 A 存在可数无限的，那么 |A|=|N|.

此外，我们将可数无限集合的基数指定为 ℵ0 (“aleph null”)。

|A|=|N|=ℵ0.

可数的

一套是可数的当且仅当它是有限的或者可数无限的。

无限的

不可数的集合是无数的或者无穷无尽。

可数集的任何子集都是可数的。

如果 S 是无限的 A⊆S 然后 A 是可数的。

超限数

如前所述，ℵ0 用来表示可数集合的基数。超限数用来描述“更高&更高”不定式的基数。

ℵ0=|N|=|Z|=|Q| 可数无限集的基数。

ℵ1=|R|=|(0,1)|=|p(N)| “最低”不可数无限集合的基数;也被称为“连续体的基数”。

ℵ2=|p(R)|=|p(p(N))| 下一个不可数无限集合的基数

由此我们可以看出 2ℵ0=ℵ1。

其他奇怪的数学可以用超限数来完成，例如 ℵ1+ℵ0=ℵ1.

集合不能映射到它的幂集合的证明类似于罗素悖论，以伯特兰·罗素命名。

这连续统假设是这样一种说法，即没有基数严格介于 N and R。连续统假说实际上是从连续统开始的推测，直到它被证明与实数系的通常公理一致(由库尔特·哥德尔在1940年提出)，并且独立于那些公理(由保罗·寇恩在1963年提出)。

统计分支

描述性统计

描述统计学是统计学的一个分支，用于通过图表或其他统计方法来描述数据。当我们有很多数据并想适当地总结它时，这就是我们做的统计类型。

推理统计

汇总数据

有多种方法可以总结数据集:

分配表

原始数据图表

样本统计，如平均值、中值、模式、标准误差和标准偏差

基于平均值的图表，带有误差线，表示标准误差或标准偏差

简单描述性统计

描述性统计是描述一组数据的数字和过程。最常见的描述性统计侧重于确定数据的“平均值”。然而，有不止一个“平均值”，所以我们在找到它们时必须具体。描述“平均值”的值有:

中值-最中间的数据，当所有数据按数量排列时

模式-集合中最常见的数据

可以想象，数据集很少都是一个值。因此，我们需要描述数据是如何围绕平均值排列的。描述平均值周围数据变化的值有:

标准差描述的是，对于整个人群，整个人群数据集的离散度。

标准误差描述了代表性样本数据集的标准偏差。

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.

上面这些数学和统计知识都是学姐千辛万苦在找来的呦，大家可能觉得学姐的这些知识点太简单了，还想要更多，希望可以给出更加专业，更加有针对性的内容，可是学姐真的能力有限，美国留学生辅导相关的深入专业知识，找留求艺的老师来帮忙就可以啦，感谢大家对学姐的支持，祝同学们早日完成自己的学业。