使用方法：

[muhat, sigmahat] = normfit(data)

[muhat, sigmahat, muci, sigmaci] = normfit(data)

[muhat, sigmahat, muci, sigmaci] = normfit(data, alpha)

[...] = normfit(data, alpha, censoring)

[...] = normfit(data, alpha, censoring, freq)

[...] = normfit(data, alpha, censoring, freq, options)

说明：

[muhat, sigmahat] = normfit(data) 返回给定数据data的正态分布均值μ 和标准差σ

的参数估计。

[muhat, sigmahat, muci, sigmaci] = normfit(data, alpha)

返回置信度区间为100(1 - alpha) % 的参数估计，这里alpha 是一个[0

1]范围内的值，由置信区间宽度决定。默认alpha 是0.05，对应95%的置信度区间。

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matlab中normfit在正态分布中的使用技巧如下:

函数 normfit

格式 [muhat,sigmahat,muci,sigmaci] = normfit(X)

[muhat,sigmahat,muci,sigmaci] = normfit(X,alpha)

说明

muhat,sigmahat分别为正态分布的参数μ和σ的估计值,muci,sigmaci分别为置信区间,其置信度为;alpha给出显著水平α,缺省时默认为0.05,即置信度为95%.

例4-62 有两组(每组100个元素)正态随机数据,其均值为10,均方差为2,求95%的置信区间和参数估计值.

解:>>r = normrnd (10,2,100,2);

%产生两列正态随机数据

>>[mu,sigma,muci,sigmaci] =

normfit(r)

则结果为

mu =

10.1455 10.0527 %各列的均值的估计值

sigma =

1.9072 2.1256 %各列的均方差的估计值

muci =

9.7652 9.6288

10.5258 10.4766

sigmaci =

1.6745 1.8663

2.2155 2.4693

说明 muci,sigmaci中各列分别为原随机数据各列估计值的置信区间,置信度为95%.

例4-63 分别使用金球和铂球测定引力常数

(1)用金球测定观察值为:6.683 6.681 6.676 6.678 6.679 6.672

(2)用铂球测定观察值为:6.661 6.661 6.667 6.667 6.664

设测定值总体为,μ和σ为未知.对(1),(2)两种情况分别求μ和σ的置信度为0.9的置信区间.

解:建立M文件:LX0833.m

X=[6.683 6.681 6.676 6.678 6.679 6.672];

Y=[6.661 6.661 6.667 6.667 6.664];

[mu,sigma,muci,sigmaci]=normfit(X,0.1) %金球测定的估计

[MU,SIGMA,MUCI,SIGMACI]=normfit(Y,0.1) %铂球测定的估计

运行后结果显示如下:

mu =

6.6782

sigma =

0.0039

muci =

6.6750

6.6813

sigmaci =

0.0026

0.0081

MU =

6.6640

SIGMA =

0.0030

MUCI =

6.6611

6.6669

SIGMACI =

0.0019

0.0071

由上可知,金球测定的μ估计值为6.6782,置信区间为[6.6750,6.6813];

σ的估计值为0.0039,置信区间为[0.0026,0.0081].

泊球测定的μ估计值为6.6640,置信区间为[6.6611,6.6669];

σ的估计值为0.0030,置信区间为[0.0019,0.0071].

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而normpdf是求概率密度函数

normpdf - Normal probability density

function

Syntax

Y = normpdf(X,mu,sigma)

Y = normpdf(X)

Y = normpdf(X,mu)

Description

Y =

normpdf(X,mu,sigma) computes the pdf at each of the values in

X using the normal distribution with mean mu and

standard deviation sigma. X, mu, and

sigma can be vectors, matrices, or multidimensional arrays

that all have the same size. A scalar input is expanded to a

constant array with the same dimensions as the other inputs. The

parameters in sigma must be positive.

The normal pdf is

The likelihood function is the pdf viewed

as a function of the parameters. Maximum likelihood estimators

(MLEs) are the values of the parameters that maximize the

likelihood function for a fixed value of x.

The standard normal distribution has

μ = 0 and σ = 1.

If x is standard normal, then

xσ + μ is also normal with mean

μ and standard deviation σ. Conversely, if

y is normal with mean μ and standard

deviation σ, then x =

(y-μ) / σ

is standard normal.

Y = normpdf(X) uses the standard normal distribution

(mu = 0,

sigma = 1).

Y = normpdf(X,mu) uses the normal distribution with

unit standard deviation

(sigma = 1).

Examples

mu = [0:0.1:2];

[y i] = max(normpdf(1.5,mu,1));

MLE = mu(i)

MLE =

1.5000