6.4.1 Common probability distributions: basic characterization of random variables 常见概率分布：随机变量的基本刻画在线视频

大家好 欢迎回到
Hello, everyone. Welcome back to

轻松学统计的课堂
the Easy Learning Statistics Class

这一讲我们要给大家介绍几种
In this lecture, I’m going to introduce you a few

常见的概率分布
common probability distributions

比如说正态分布
Let's say normal distribution

t分布 F分布等等
t distribution, F distribution, and so on

首先我们来介绍正态分布
Let's start with normal distribution

正态分布可能是大家最为熟悉的
Normal distribution is probably a distribution

一种分布
You’re most familiar with

在中学阶段
In middle school

可能大家就接触过
you might contact with

正态分布
normal distribution

最初是由高斯作为描述
It was originally presented by Gauss for describing

误差相对频数分布的模型而提出的
model of the relative frequency distribution of errors

但是令人惊讶的是
But what is surprising is that

这条曲线竟然为许多不同领域的
this curve is actually, for the relative frequency of data

数据的相对频数
in many fields

提供了一个恰当的模型
provides an appropriate model

因而得到了十分广泛的应用
So it has been widely used

正态分布在概率论与数理统计当中
in probability theory and mathematical statistics, normal distribution

具有非常重要的地位
is of great importance

在现实生活当中有许多现象
There are many phenomena in real life

都可以由正态分布来描述
can be described by normal distribution

其它的一些分布
Some of the other distributions

也可以利用正态分布做近似计算
can use normal distribution for approximate calculation

而且正态分布也可以导出
And normal distribution can be used to derive

一些比如说卡方分布
something like chi-square distribution

t分布 F分布等等
t distribution, F distribution, and so on

那么正态分布比如说
In respect o
f normal distribution, for example

我们的身高 体重 智商
our height, weight and IQ

再比如说我们体检当中的
and in our physical examination

红细胞数 血红蛋白量这些
red blood cell count, hemoglobin, etc.

这都服从正态分布
They all obey normal distribution

它的应用是非常广泛的
Its application is very extensive

接下来大家看一下
Then, let’s see

这就是正态分布的概率分布
This is the probability distribution density function of

密度函数
density function

那么它有两个非常重要的参数
It has two very important parameters

一个是μ
One is μ

就是平均数
or average

一个是σ平方就是方差
The other is the square of σ，which is the variance

那么如果一个随机变量
If a random variable

服从这样的概率分布密度函数的话
obeys such probability distribution density function

我们就称这个随机变量
we say this random variable

服从正态分布
obeys normal distribution

那么大家可以看到
We can see

我们这个正态分布的这个
the curves of probability densities

概率密度的曲线
of normal distribution

那么这个不同形状的
These density function curves

不同位置的这些概率密度
have different shapes

函数的曲线
at different positions

它主要是因为μ和σ平方的不同
It is the difference in square between μ and σ

造成的它这样不同的一个
that leads to such different

曲线的这样一个形态
patterns of curves

接下来
Next

大家看到的这张图
The graph you see

就是正态分布的这个分布函数
shows the distribution function of normal distribution

那么怎么样从这个密度函数
How do we derive the distribution function

推出分布函数
from this density function

我想这个是大家在
I think this is

概率论与数理统计当中
a content you are familiar

比较熟悉的内容
in probability theory and mathematical statistics

正态分布的特性主要有
The characteristics of the normal distribution are mainly that

它的密度函数的曲线
the curve of its density function

是一个单峰 对称的钟形的曲线
is a bell-shaped curve with a single peak

它的对称轴是均值
Its axis of symmetry is the average

也就是X=μ
That is, X=μ

那么密度函数的曲线
The curve of density function

在X=μ的时候
when X=μ

达到最大值
reaches the maximum value

那么正态分布的密度函数
The density function of normal distribution

也是一个非负的函数
is also a non-negative function

渐近线是X轴
The asymptote is the X-axis

也就是说它的取值范围
That is to say its value range

是负无穷到正无穷
is from negative infinity to positive infinity

另外密度函数的曲线
Besides, the area included between the density function curve

与横轴所夹的面积为1
and the horizontal axis is 1

以上就是我们经常会用到的
The above are some properties of

正态分布密度函数曲线的一些性质
normal distribution density function curve that we will frequently use

接下来我们经常还要做的一件事
Next, we will see

就是把一个服从一般正态分布的
how to transform a random variable obeying

随机变量
general normal distribution

转化成服从标准正态分布
into one obeying standard normal distribution

应该怎么做
How should we do

比如说有一个随机变量X
Let's say there is a random variable, X

它服从均值为μ
which obeys a normal distribution with

方差为σ平方的正态分布
average of μ and variance of σ squared

那么我们要标准化
Then, we’ll standardize it

就是（公式如上）
(The formula is as above)

得到一个新的随机变量Z
to get a new random variable, Z

那么这个新的随机变量Z
Then, this new random variable, Z

它就是服从均值为0
is a standard normal distribution

方差为1的标准正态分布
with average 0 and variance 1

那么从一般正态分布到标准正态分布
From a normal distribution to a standard normal distribution

这个过程是大家要熟悉的
this is a process you should get familiar with

那么有些同学可能会问
Some students may ask

我们为什么要标准化呢
why should we get it standardized

主要是因为
This is mainly because that

我们正态分布它是一个分布族
the normal distribution is a family of distribution

那么不同的均值
Under different average values

不同的方差下
and different variances

那么我们正态分布的密度函数曲线
the density function curves of normal distribution

就有不同的形态
can have different patterns

那么我们在每一种形态下去做概率
If we do probability of every pattern

那么整个过程就会比较复杂
the process can be complex

如果我们能把一般的正态分布
If we can transform general normal distributions

都能转化成同一种分布
into a same type of distribution

也就是我们的标准正态分布
that is, standard normal distribution

那么我们计算概率的工作
the workload of probability calculation

就大大简化了
can be greatly simplified

那么为什么在
Then, why

概率论与这个数理统计当中
in probability theory and mathematical statistics

正态分布这么重要
normal distribution is so important

我们说主要的原因有以下的三点
There are three main reasons for this

第一我们说它是最常见的
First, we say it's the most common

刚才我们也给大家看了一些例子
We just showed you some examples

那么非常多的现象
Many phenomena

都是服从
all obey

或者近似服从正态分布的
or approximately obey normal distribution

那么第二个原因就是
The second reason is

在一定的条件下
under certain conditions

正态分布是其他分布的近似分布
the normal distribution is an approximate distribution of other distributions

比如说我们说当n很大的时候
Let's say that when n is large

我们的t分布就趋近于正态分布
So our t-distribution tends to be a normal distribution

第三个方面主要在于正态分布
The third reason is that from the normal distribution

我们说可以推导出来很多有用的分布
we can derive a lot of useful distributions

比如说我们小样本的精确分布
For example, the exact distribution of our small sample

就是由正态分布推导出来的
is just derived from normal distribution

这就是在正态分布这个部分
These are the contents in normal distribution

我们需要回顾的内容
that we need to review

接下来我们给大家介绍的卡方分布
Now let's introduce you to the chi-square distribution

卡方分布是由阿贝
The chi-square distribution was first proposed by Abbe

在1863年首先提出来的
in 1863

后来又由海尔墨特和卡尔・皮尔逊
and it was deduced by Hermert and C.K. Pearson

在1875和1900年推导出来
in 1875 and 1900, respectively

那么他的过程大家也来看一下
Let’s see the process

如果X服从一个一般的正态分布
If X follows a normal distribution

均值为μ方差为σ平方
and the average is μ and the variance is σ squared

那么我们就会得到
we’ll obtain

刚才我们说标准化的过程
As in the standardization process, we talked just now

会得到一个新的随机变量
a new random variable

（公式如上）
(The formula is as above)

也就是我们的Z这个随机变量
That's our random variable Z

它就服从标准正态分布
obeys the standard normal distribution

那么这个时候
So at this point

服从标准正态分布的随机变量的平方
the square of the random variable that obeys the standard normal distribution

它就服从自由度为1的卡方分布
follows the chi-square distribution with a degree of freedom of 1

那么我们这样来做一个标记
So let's make a notation like this

那么卡方分布的一个
One of the very important features of

非常重要的特征就是
Chi-square distributions is

如果随机变量X1 X2到Xn
if the random variables X1, X2 up to Xn

都是相互独立的
are all independent of each other

并且都服从标准正态分布的话
and all follow standard normal distribution

那么它们的平方和就会服从
the sum of their squares will obey

自由度为n的卡方分布
the chi-square distribution with a degree of freedom of n

这是卡方分布常用的一个
This is one of the commonly used mathematical properties

数学的性质
of chi-square distribution

接下来我们来看一下
So let's look at

卡方分布的这个性质和特点
the properties and characteristic of the chi-square distribution

第一我们说卡方分布的变量值
First, the variable values of the chi-square distribution

始终为正
are always positive

其实就是说它的这个概率
It's essentially saying that

函数密度的这个曲线
the curves of the probability density functions

始终在第一象限
are always in the first quadrant

第二 卡方分布它分布的这个形状
Second, the distribution shape of the chi-square distribution

我们说跟它的自由度
has a close relation with

是有密切的关系的
its degree of freedom

通常是不对称的一个正偏分布
It's usually an asymmetric positively skewed distribution

但是随着自由度的增大
But, with the increase in the degree of freedom

我们说她会趋于对称
it will tend to be symmetrical

另外我们经常会用到的一个特性就是
And one of the other features frequently used is

服从卡方分布的这个随机变量
this random variable that obeys the chi-square distribution

它的均值就是n
Its average is n

n其实就是我们刚才说的
N is actually what we just said

它的自由度
its degree of freedom

它的方差是2倍的n
Its variance is 2 times n

那么刚才我们也提到了这个卡方分布
We also mentioned that this chi-square distribution

它具有一个可加性
has additivity

就是如果U和V是两个独立的
If U and V are independent

都服从卡方分布的随机变量
and both follow the random variable of the chi-square distribution

那么U＋V它也服从卡方分布
So U + V also follows the chi-square distribution

服从自由度为n1＋n2的卡方分布
follows the chi-square distribution with a degree of freedom of n1＋n2

这个性质是用的比较多的
This property is much used

接下来大家可以看到卡方分布的
Now, let’s look at the schematic diagram

这个示意图
of chi-square distribution

就是它的概率 密度 函数曲线的
That is the basic characteristic of

这样一个基本的一个特征
its probability density function curve

那么大家也可以看到
You can see that

随着自由度的增大
with the increase of the degree of freedom

它越来越趋于对称
it becomes more and more symmetrical

再接下来我们再来介绍一种
Next, we’ll introduce another

常见的分布
common distribution

就是t分布
That is t distribution

那么t分布我们也被称其为
t distribution is also called

学生式分布
student’s distribution

是由这个哥赛特在1908年
It was first proposed by

首次提出来的
Gosset in 1988

它的重要意义在于
The significance of this is that

它提供了小样本的研究方法
it provides a method for studying small samples

那么它是怎么构造的呢
Then, how is it constructed

我们说如果随机变量X
If a random variable X

服从标准正态分布
obeys the standard normal distribution

随机变量Y
a random variable Y

服从自由度为n的卡方分布
follows the chi-square distribution with a degree of freedom of n

且X Y相互独立
and X and Y are independent of each other

那么我们就可以构造一个
then we can construct

服从t分布的随机变量
a random variable that obeys t-distribution

我们就用（公式如上）
So let's just use the above formula

那么这样的话
In this case

我们得到这样的一个随机变量
we get a random variable that

它就服从自由度为n的t分布
follows the t distribution with n degree of freedom

这个就是t分布的
This is an initial structure

最初的这样一个构造
of t-distribution

那么接下来大家可以看一下
Next, you can look at

t分布的这个密度函数的图象
the graph of the density function of the t-distribution

我们从t分布的密度函数曲线当中
From the density function curve of the t-distribution

可以看到
it can been seen that

t分布它是关于Y轴对称的
the t-distribution is symmetric about the Y-axis

那么还有一个比较重要的特点
It has another important characteristic

就是随着自由度的增大
With the increase of the degree of freedom

t分布的密度函数曲线
the density function curve of the t-distribution

逐渐逼近标准正态分布
gradually approaches the standard normal distribution

这个是一个非常重要的特性
This is a very important feature

再接下来我们说我们会看到F分布
Next, we’ll see F distribution

F分布我们说是由
F-distribution was initially proposed by

统计学家费希尔最初提出的
statistician Fisher

那么以他姓氏的第一个字母来命名的
and named after the first letter of his last name

那么F分布它是怎么构造的呢
How is the F distribution constructed

我们来看一下
Let’s have a look

如果U服从自由度为n1的卡方分布
If U follows the chi-square distribution with n1 degree of freedom

V服从自由度为n2的卡方分布
and V follows a chi-square distribution of n2 degree of freedom

那么我们就在U和V
then, under the premise that

相互独立的这个前提下
U and V are independent of each other

那么我们就可以构造一个F
Then we can construct

服从F分布的一个随机变量
a random variable that obeys the F distribution

怎么构造的呢
How to construct it

（公式如上）
(The formula is as above)

那么这个时候构造的这个
The random variable F

随机变量F
we constructed at this time

它就服从自由度
obeys an F distribution that has

第一自由度为n1
a first degree of freedom of n1

第二自由度为n2的F分布
and the second degree of freedom of n2

这就是我们的F分布的
This is the F-distribution’s

最初的这个构造
initial structure

下面我们也可以看一下
Now, let’s have a look at

在不同自由度下
the density function curves of F distribution

F分布的密度函数的曲线
under different degrees of freedom

那么大家也可以看到
you can see that

F分布的密度函数的曲线
the density function curves of F distribution

也是在第一象限
are also in the first quadrant

并且随着自由度的这个变化
and with changes in degrees of freedom

它的曲线的形态
the patterns of the curves

我们说也有相应的变化
change accordingly

随着自由度的增大
As the degrees of freedom increase

F分布的密度曲线越来越陡
the e density curves of the F distribution get steeper and steeper

那么并且我们说
And

当自由度
when the degrees of freedom

它的第一自由度
its first degree of freedom

和它的第二自由度
and its second degree of freedom

都趋近于无穷大的时候
both approach infinite

F分布的密度函数的曲线
the density curves of the F distribution

近似的关于X＝1对称
are approximately symmetric with respect to X = 1

这个是大家需要注意的
This is the characteristic of the F distribution

F分布的特性
you should pay attention to