title: 正态分布与参数估计
categories:
- 杂项
tags: - 学习
正态分布与参数估计
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