问题 设总体概率函数如下, $x_1,x_2,\cdots,x_n$是样本,试求未知参数的最大似然估计. 1. $p(x;\theta)=c\theta^c x^{-(r+1)}, x>\theta, \theta>0, c>0$已知； 2. $p(x;\theta,\mu)=\frac{1}{\theta}e^{-\frac{x-\mu}{\theta}}, x>\mu, \theta>0$; 3. $p(x;\theta)=(k\theta)^{-1}, \theta0, k>0$已知. ## 答案 (1) 样本 $x_1, x_2, \ldots, x_n$ 的似然函数为 $$ L(\theta) = \prod_{i=1}^n p(x_i; \theta) = \prod_{i=1}^n \left[ c \theta^{c} x_i^{-(c+1)} , I_{{ x_i > \theta }} \right] = c^{n} \theta^{n c} \left( \prod_{i=1}^n x_i^{-(c+1)} \right) I_{{ \theta

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