Question

X,Y独立，为什么X+Y~N（1,2）能推出P（X+Y≤1）=1/2?

Answer 1

正态分布以x=μ为对称轴，μ表示其均值，很显然落在对称轴左右两边的概率各位1/2，这也就是式子的几何意义

Answer 2

答：π (e - 1) 极坐标化简 x = rcosθ y = rsinθ x2+y2=r2,0≤r≤1,0≤θ≤2π ∫∫_(D) e^(x2+y2) dxdy = ∫(0,2π) dθ ∫(0,1) e^r2 * r dr = (2π)∫(0,1) e^r2 d(r2)/2 = π * [e^r2](0,1) = π * (e^1 - e^0) = π (e - 1)