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Matlab累积分布函数cdf与概率密度函数pdf

Matlab累积分布函数cdf与概率密度函数pdf

作者: 来自外星球的柠檬甜橙 | 来源:发表于2019-05-09 12:33 被阅读0次

原文:http://blog.sina.com.cn/s/blog_7054a1960102vy7x.html

累积分布函数cdf (Cumulative Distribution Function)

背景知识:http://www.lifelaf.com/blog/?p=746

语法

y = cdf('name',x,A)

y = cdf('name',x,A,B)

y = cdf('name',x,A,B,C)

y = cdf(pd,x)

y = cdf(___,'upper')

描述

y = cdf('name',x,A) 计算某种分布(由'name'定义,如'Normal'正态, 'Poisson'泊松, 'T' t分布…)下,x值处的累计分布,A,B,C等为'name'函数的参数

y = cdf(pd,x) 直接计算概率分布函数pd(probability distribution) ,在x处的累计分布,实际上,这里的pd 已被'name', A定义好,举栗如下:

% 定义一个正态分布函数pd, 均值mu = 0, 标准差sigma = 1.

mu = 0;

sigma = 1;

pd = makedist('Normal',mu,sigma);

% 定义x

x = [-2,-1,0,1,2];

% 计算x值处的累计分布

y = cdf(pd,x)

y =

 0.0228 0.1587 0.5000 0.8413 0.9772

用第一种语句表达相同内容为:

y2 = cdf('Normal',x,mu,sigma) %正态分布,x值处,均值为0,标准差为1

y2 =

 0.0228 0.1587 0.5000 0.8413 0.9772

http://nl.mathworks.com/help/stats/cdf.html

t分布累积分布函数tcdf (Student'stcumulative distribution function)

% 事实上就是y = cdf('T',x,A)函数

语法

p = tcdf(x,nu)

p = tcdf(x,nu,'upper')

描述

计算t分布在x值处的累积分布,nu是t分布的自由度

再举个栗子

mu = 1; % Population mean

sigma = 2;% Population standard deviation

n = 100; % Sample size

x = normrnd(mu,sigma,n,1);% Random sample from population

xbar = mean(x);% Sample mean

s = std(x); % Sample standard deviation

t = (xbar - mu)/(s/sqrt(n)) % 这里t分布出现了,正态分布总体与样本均值的差符合t分布

t =

 1.0589

p = 1-tcdf(t,n-1) % Probability of larger t-statistic

p =

 0.1461

该p值(即t函数的累积分布就是t检验在相同x值处的概率ptest)

[h,ptest] = ttest(x,mu,0.05,'right')

h =

 0

ptest =

 0.1461

http://nl.mathworks.com/help/stats/tcdf.html

概率密度函数pdf (Probability density functions)

搞懂了累积分布函数cdf,这个就没什么需要多说了

语法

y = pdf('name',x,A)

y = pdf('name',x,A,B)

y = pdf('name',x,A,B,C)

y = pdf(pd,x)

举例

% 定义一个正态分布函数pd, 均值mu = 0, 标准差sigma = 1.

mu = 0;

sigma = 1;

pd = makedist('Normal',mu,sigma);

% 定义x值

x = [-2 -1 0 1 2];

% 计算x值处的概率密度(cdf是累计分布)

y = pdf (pd,x)

y =

0.0540 0.2420 0.3989 0.2420 0.0540

同样,另一种表达

y = pdf(pd,x)

y =

 0.0540 0.2420 0.3989 0.2420 0.0540

http://nl.mathworks.com/help/stats/pdf.html

t分布概率密度函数tpdfStudent's t probability density function

语法

y = tpdf(x,nu)

举例

tpdf(0,1:6)

ans =

 0.3183 0.3536 0.3676 0.3750 0.3796 0.3827

http://nl.mathworks.com/help/stats/tpdf.html

相反,还可以通过p求t分布的t值

tinv (Student's t inverse cumulative distribution function)

语法

x = tinv(p,nu)

举例

% the 99th percentile of the Student's t distribution for one to six degrees of freedom

percentile = tinv(0.99,1:6)

percentile =

 31.8205 6.9646 4.5407 3.7469 3.3649 3.1427

http://nl.mathworks.com/help/stats/tinv.html

有一个问题,Matlab有一个inv矩阵求逆函数,不知与tinv什么关系,莫非tinv是在t分布下调用了inv计算程序?但p并不等是t的逆矩阵啊(即t*p = E)啊?求解答

inv是矩阵求逆的意思。具体用法A=inv(B),其中B是输入的可逆矩阵,输出A就是B的逆矩阵,逆矩阵满足性质 AB=BA=E (E是单位阵)。如果输入的是不可逆矩阵会弹出警告,并返回inf。

调用举例:

>> inv([1 0;0 0])

警告: 矩阵为奇异工作精度。

ans =

Inf Inf

Inf Inf

>> inv(rand(2))

ans =

-13.0929 5.2640

12.0501 -3.3159

另附官方英文解释(输入doc inv也可以自己查看):

Y = inv(X) returns theinverse of the square matrix X. A warning messageis printed if X is badly scaled or nearly singular.

In practice, it is seldom necessary to form the explicit inverseof a matrix. A frequent misuse of inv arises whensolving the system of linear equations Ax = b.One way to solve this is with x = inv(A)*b.A better way, from both an execution time and numerical accuracy standpoint,is to use the matrix division operator x = A\b.This produces the solution using Gaussian elimination, without formingthe inverse. See mldivide (\)for further information.

http://zhidao.baidu.com/link?url=dTjjN7fsj2EBX7zwCm1_TZA2LDv4Abtmgnq0mwfb3pyCLpZ26g1RWFGFlKOiADzVsnpeUh-bu4o9QvP_e5T5q

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