Mathematics · Statistics

MATH2099 · 抽样分布与 CLT

抽样分布与 CLT

~ 10 min · v1

抽样分布与 CLT

和 ~ N(nμ,nσ²)、均值 ~ N(μ,σ²/n) · Lecture 4

虾总质量 (CLT 和)

S=\sum_{i=1}^{n}X_i\ \sim\ N(n\mu,\ n\sigma^2),\quad \sigma_S=\sqrt{n}\,\sigma

电梯载重 (CLT 和)

S\sim N(n\mu,\ n\sigma^2)\qquad \ell=n\mu+z_{1-\alpha}\sqrt{n}\,\sigma

日产量 (CLT 均值)

\bar{X}\sim N\!\left(\mu,\frac{\sigma^2}{n}\right)\qquad \ell=\mu+z_{\alpha}\frac{\sigma}{\sqrt{n}}

螺栓批均重 (双侧 + 反解 n)

P(\text{out})=2\left[1-\Phi\!\left(\frac{d}{\sigma/\sqrt{n}}\right)\right] \qquad n=\left(\frac{z_{1-\alpha/2}\,\sigma}{d}\right)^{2}

树高 (CLT 均值 + 反解 n)

\left(\frac{c-\mu}{\sigma/\sqrt{n}}\right)=z_{\alpha}\ \Longrightarrow\ n=\left(\frac{z_{\alpha}\,\sigma}{c-\mu}\right)^{2}\ \ \text{(向上取整)}