Nuclear Astrophysics , 2025 Update
摘要
For the sake of ready reference, the basic materials will be restated here. Let \(x_1>0\) and \(x_2>0\) be two real scalar positive variables with the associated functions \(f_1(x_1)\) and \(f_2(x_2)\) respectively. Let the joint function of \(x_1\) and \(x_2\) be \(f_1(x_1)f_2(x_2)\) , the product. If \(x_1>0\) and \(x_2>0\) are real scalar random variables with the densities \(f_1(x_1)\) and \(f_2(x_2)\) , then we say that \(x_1\) and \(x_2\) are statistically independently distributed when we take the joint density as \(f_1(x_1)f_2(x_2)\) , the product. Let \(u=x_1x_2\) the product. Consider the transformation \(u=x_1x_2,v=x_2\) . Then, we can see that the wedge product of differentials are connected by the relation \(\textrm{d}x_1\wedge \textrm{d}x_2=\frac{1}{v}\textrm{d}u\wedge \textrm{d}v\) and then the marginal function of u, denoted by g(u), is given by the following: