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,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值

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 =

0.0228 0.1587 0.5000 0.8413 0.9772

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

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

t分布概率密度函数tpdf(Student'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

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

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.

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,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值

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 =

0.0228 0.1587 0.5000 0.8413 0.9772

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

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

t分布概率密度函数tpdf(Student'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

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

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 =

Ab.This produces the solution using Gaussian elimination, without

formingthe inverse. See mldivide ()for further information.

cdf

Y = cdf(name,X,A)

Y = cdf(name,X,A,B)

Y = cdf(name,X,A,B,C)

Description

Y = cdf(name,X,A) computes the cumulative distribution function for the one-parameter family of distributions specified by name. A contains parameter values for the distribution. The cumulative distribution function is evaluated at the values in X and its values are returned in Y.

If X and A are arrays, they must be the same size. If X is a scalar, it is expanded to a constant matrix the same size as A. If A is a scalar, it is expanded to a constant matrix the same size as X.

Y is the common size of X and A after any necessary scalar expansion.

Y = cdf(name,X,A,B) computes the cumulative distribution function for two-parameter families of distributions, where parameter values are given in A and B.

If X, A, and B are arrays, they must be the same size. If X is a scalar, it is expanded to a constant matrix the same size as A and B. If either A or B are scalars, they are expanded to constant matrices the same size as X.

Y is the common size of X, A, and B after any necessary scalar expansion.

Y = cdf(name,X,A,B,C) computes the cumulative distribution function for three-parameter families of distributions, where parameter values are given in A, B, and C.

If X, A, B, and C are arrays, they must be the same size. If X is a scalar, it is expanded to a constant matrix the same size as A, B, and C. If any of A, B or C are scalars, they are expanded to constant matrices the same size as X.

Y is the common size of X, A, B, and C after any necessary scalar expansion.

Acceptable strings for name are:

*     'beta' (Beta distribution)

*      'bino' (Binomial distribution)

*      'chi2' (Chi-square distribution)

*      'exp' (Exponential distribution)

*      'ev' (Extreme value distribution)

*      'f' (F distribution)

*      'gam' (Gamma distribution)

*      'gev' (Generalized extreme value distribution)

*      'gp' (Generalized Pareto distribution)

*      'geo' (Geometric distribution)

*      'hyge' (Hypergeometric distribution)

*      'logn' (Lognormal distribution)

*      'nbin' (Negative binomial distribution)

*      'ncf' (Noncentral F distribution)

*      'nct' (Noncentral tdistribution)

*      'ncx2' (Noncentral chi-square distribution)

*      'norm' (Normal distribution)

*      'poiss' (Poisson distribution)

*      'rayl' (Rayleigh distribution)

*      't' (t distribution)

*      'unif' (Uniform distribution)

*      'unid' (Discrete uniform distribution)

*      'wbl' (Weibull distribution)