在概率论中,求解形如E[φ(X)] -1/(√2πσ)∫-∞+∞φ(x)e-(x-μ)2/(2σ2)ds的积分是很重要的.但即使φ(x)是初等函数如xn,ems,sinmx等,用常规的分部积分法也不易处理.而1维热传导方程初值问题有形如前面的积分解和含有微分算子的级数解,由解的唯一性将把这类期望的积分运算转化为含有微分的级数运算.通过举例说明了该方法在求解数字特征、特征函数等方面的简便实用性,并以公式形式给出了xn,ems,sinmx等解析函数的期望.最后作为补充,给出了n维类似的结论.%In probability theory, the integral such asE [ψ(X) ] =1 -Γ2πσ∫+∞-∞ ψ(x)e-(x-n)2-2σ2 dx is very important,but it is not easy to solve it by using integration by parts, though ψ( x) is the elementary function such asxn, emx , smmx,etc. The solution of the Cauchy problem of one-dimension heat equation has two forms: the integral as E[ψ> (X) ] which was mentioned and the progression with differential operator. Because of the unique of the solution, integral oper-ation can be changed into differential operation. The formulas will be easily given in the form of examples about n-umerical characteristic and characteristic function, etc, when ψ( x) is the elementary function such as xn, emx, sinmx, and the product of them. Finally, the theory of re-dimension heat equation will be given.
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