摘要： {\bf Question 2:\ }Suppose that $X$ has density function\[f_\theta(x) = \frac{x^{\theta-1} e^{-x}}{\Gamma(\theta)} I \{ x>0\}.\]Find expressions for t... 阅读全文

摘要： {\bf Question 1:}Let $X\sim f_\theta(x)$ for some density function $f_\theta(x)$ and define\[I(\theta) = E_\theta\left[ \left( \frac{\partial}{\partia... 阅读全文

摘要： R code for generating standard normals using Metropolis sampler with uniform proposal distribution:# metropolis for N(0,1) based on uniform candidates... 阅读全文

摘要： How can I prove $$\int[F(x+a)-F(x)]\,dx=a$$where $F(x)$ is the cumulative distribution function?Proof:Let $R, S> 0$ be large compared to $a$. Then$$\b... 阅读全文

摘要： 证明积分：$$\int_{-\pi/2}^{\pi/2} (\sin(x))^n dx = \frac{n-1}{n}\int_{-\pi/2}^{\pi/2} (\sin(x))^{n-2} dx$$证明：\begin{align}\int_{-\pi/2}^{\pi/2} \sin^nx \, ... 阅读全文

摘要： 计算$$\int_0^{+\infty} \dfrac{1}{1+x^6}\,dx$$解答:本题可以利用一个结论$$\int_0^{+\infty}\dfrac{x^{p-1}}{1+x}\,dx=\Gamma(p)\Gamma(1-p)=\dfrac{\pi}{\sin(\pi p)},\quad 00$.所以有$$\int_{0}^{+\infty }{\frac{1}{1+{{x}^{6}}}dx}=\frac{1}{2}\int_{-\infty }^{+\infty }{\frac{1}{1+{{x}^{6}}}dx}=\pi i\sum\limits_{\operatorname{ 阅读全文

摘要： $$\int_a^b f(x)\,dx$$博客园支持Latex的公式如下:http://www.cnblogs.com/cmt/p/3279312.html 阅读全文