例 6.1.1 (正态分布的样本联合分布)

总体 $X\sim N\left(\mu ,{\sigma }^2\right),\left(X_1,X_2,\cdots ,X_n\right)$ 为其样本, 则 $(X_1,X_2,\cdots$, $X_n)$ 的联合分布密度函数为

$$\begin{array}{rl}& f_{X_1,X_2,\cdots ,X_n}\left(x_1,x_2,\cdots ,x_n;\mu ,{\sigma }^2\right)\\ & =f_X\left(x_1,\mu ,{\sigma }^2\right)f_X\left(x_2,\mu ,{\sigma }^2\right)\cdots f_X\left(x_n,\mu ,{\sigma }^2\right)\\ & =\frac{1}{\sqrt{2\pi }\sigma }{\mathrm{e}}^{-\frac{{\left(x_1-\mu \right)}^2}{2{\sigma }^2}}\cdot \frac{1}{\sqrt{2\pi }\sigma }{\mathrm{e}}^{-\frac{{\left(x_2-\mu \right)}^2}{2{\sigma }^2}}\cdots \frac{1}{\sqrt{2\pi }\sigma }{\mathrm{e}}^{-\frac{{\left(x_n-\mu \right)}^2}{2{\sigma }^2}}\\ & ={\left(2\pi {\sigma }^2\right)}^{-\frac{n}{2}}\exp \left(-\frac{1}{2{\sigma }^2}\sum _{i=1}^n{\left(x_i-\mu \right)}^2\right).\end{array}$$