For normal distribution N(0, 1), find a number z* such that P(−z∗<Z<z∗)=0.05
时间: 2024-02-17 20:05:22 浏览: 18
根据题意,需要找到一个数 $z^*$,使得 $P(-z^*<Z<z^*)=0.05$,其中 $Z$ 是标准正态分布。
由于标准正态分布是关于 $0$ 对称的,即 $P(Z<-z^*)=P(Z>z^*)$,因此可以转化为求 $P(Z>z^*)$ 为 $0.025$,即:
$$P(Z>z^*)=0.025$$
根据正态分布的对称性,$P(Z<-z^*)=0.025$。
可以使用R语言中的`qnorm()`函数计算出 $z^*$ 的值:
```
qnorm(0.025, lower.tail = FALSE)
```
得到的结果为 $1.959964$(保留小数点后四位)。
因此,$z^*=1.96$(取到小数点后两位即可)。所以 $P(-1.96<Z<1.96)=0.05$。
相关问题
用matlab编程Generate a candidate draw θ* from the normal proposal distribution θ ˜∗ = θ ˜(s−1) + z, z ∼ N (0, Σ) 。 where Σ a diagonal matrix, whose diagonal elements are the diagonal elements of the inverse of the negative Hessian matrix of ˜p(θ) evaluated at θ (0), i.e., Σ = diag − ∂p˜(θ) 2 ∂θ∂θ′ θ=θ0 ! −1
Sure, here's an example code in MATLAB:
```
% Define the proposal distribution
sigma = 2; % standard deviation
Sigma = diag([1/sigma^2 1/sigma^2]); % diagonal covariance matrix
% Define the negative Hessian matrix of the target distribution
H = [1 0.5; 0.5 2]; % example Hessian matrix
H_inv = inv(H); % inverse of the Hessian matrix
% Evaluate the diagonal elements of Sigma
Sigma_diag = diag(-H_inv);
% Generate a candidate draw theta_star
if s == 1 % if it's the first iteration, start from an initial value
theta_s = [0; 0]; % initial value
else % if it's not the first iteration, use the previous value as the starting point
theta_s = theta_star;
end
z = mvnrnd([0 0], Sigma); % generate z from the proposal distribution
theta_star = theta_s + z; % candidate draw
```
Note that you need to replace the example Hessian matrix (`H`) and standard deviation (`sigma`) with the actual values from your problem. Also, this code assumes that the target distribution is a bivariate normal distribution, so you may need to modify it if your target distribution is different.
用matlab进行如下操作:Generate a candidate draw θ ∗ from the normal proposal distribution θ ˜∗ = θ ˜(s−1) + z, z ∼ N (0, Σ) (26) where Σ a diagonal matrix, whose diagonal elements are the diagonal elements of the inverse of the negative Hessian matrix of ˜p(θ) evaluated at θ (0), i.e., Σ = diag − ∂p˜(θ) 2 ∂θ∂θ′ θ=θ0 ! −1
以下是 Matlab 代码实现:
假设当前状态为 theta,先计算出 negative Hessian matrix H:
```matlab
H = -hessian( theta, p_tilde );
```
其中 hessian 函数为自定义函数,用于计算 Hessian 矩阵,具体代码如下:
```matlab
function H = hessian( x, f )
% HESSIAN Computes the Hessian matrix of a function f(x).
% Usage:
% H = hessian( x, f )
% Inputs:
% x - n x 1 column vector, point where to compute the Hessian matrix
% f - function handle, f(x) should return a scalar
% Outputs:
% H - n x n matrix, the Hessian matrix of f(x) at point x
n = length(x);
H = zeros(n,n);
for i=1:n
for j=1:n
% Compute second-order partial derivatives numerically
fxx = @(x)f([x(1:i-1);x(i)+eps;x(i+1:j-1);x(j)+eps;x(j+1:n)]);
fxy = @(x)f([x(1:i-1);x(i)+eps;x(i+1:j-1);x(j)+eps;x(j+1:n)]);
fyx = @(x)f([x(1:i-1);x(i)+eps;x(i+1:j-1);x(j)+eps;x(j+1:n)]);
fyy = @(x)f([x(1:i-1);x(i)+eps;x(i+1:j-1);x(j)+eps;x(j+1:n)]);
H(i,j) = (fxx(x)-fxy(x)-fyx(x)+fyy(x)) / (4*eps^2);
end
end
end
```
计算出 H 后,再计算出 Σ:
```matlab
Sigma = diag( diag( inv(H) ) );
```
然后生成候选状态:
```matlab
z = randn( size(theta) );
theta_star = theta + Sigma*z;
```
其中 randn 函数用于生成服从标准正态分布的随机数。