正态分布与参数估计

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预处理

clear
rng(6331);
mu = 1;
sigma = 1;

真实概率密度曲线：

fplot(@(x) exp(-(x-mu).^2./(2*sigma))./(sqrt(2*pi)*sigma) )

title("真实概率密度曲线")

四个样本直方图

n_1 = 10;

n_2 = 50;

n_3 = 500;

n_4 = 1000;

y_1 = sigma.*randn(n_1,1) + mu;

y_2 = sigma.*randn(n_2,1) + mu;

y_3 = sigma.*randn(n_3,1) + mu;

y_4 = sigma.*randn(n_4,1) + mu;

histogram(y_1,10)

title("n=10")

histogram(y_2,10)

title("n=50")

histogram(y_3,20)

title("n=500")

histogram(y_4,20)

title("n=1000")

四个样本参数估计值

矩估计==最大似然估计

[mu_, sigma_] =estimate(y_1 , n_1)
mu_ = 0.6865
sigma_ = 0.2972
[mu_, sigma_] =estimate(y_2 , n_2)
mu_ = 0.7853
sigma_ = 1.0036
[mu_, sigma_] =estimate(y_3 , n_3)
mu_ = 0.9213
sigma_ = 1.0244
[mu_, sigma_] =estimate(y_4 , n_4)
mu_ = 0.9908
sigma_ = 1.0422

参数估计误差随样本容量增大的变化曲线

矩估计==最大似然估计

样本容量取 10~1000

n = 10:1:1000;

dsigma = zeros(length(n),1);

dmu = zeros(length(n),1);

for i = 1:length(n)

y = sigma.*randn(n(i),1) + mu;

[mu_, sigma_] =estimate(y , n(i));

dsigma(i) = abs(sigma - sigma_);

dmu(i) = abs(mu - mu_);

end

plot(n,dsigma)

hold on

plot(n,dmu)

legend('△σ','△μ')

hold off

矩估计、最大似然估计函数

function [mu_, sigma_] = estimate(y , n)
    mu_ = sum(y)/n;
    sigma_ = sum((y-mu_).^2)/n;
end