相互独立的随机变量分布及卷积公式

(X,Y)∼f(x,y),Z=X+Y,fZ(z)=∫−∞+∞fX(x)fY(z−x)dx=∫−∞+∞fX(z−y)fY(y)dy(X,Y) \sim f(x,y), Z=X+Y, f_Z(z)=\int_{-\infty}^{+\infty}f_X(x)f_Y(z-x)dx = \int_{-\infty}^{+\infty}f_X(z-y)f_Y(y)dy (X,Y)∼f(x,y),Z=X−Y,fZ(z)=∫−∞+∞f(x,x−z)dx=∫−∞+∞fX(z+y)fY(y)dy(X,Y) \sim f(x,y), Z=X-Y, f_Z(z)=\int_{-\infty}^{+\infty}f(x,x-z)dx = \int_{-\infty}^{+\infty}f_X(z+y)f_Y(y)dy (X,Y)∼f(x,y),Z=XY,fZ(z)=∫−∞+∞1∣x∣f(x,zx)dx=∫−∞+∞1∣y∣f(zy,y)dy(X,Y) \sim f(x,y), Z=XY, f_Z(z)=\int_{-\infty}^{+\infty} \frac{1}{|x|} f(x, \frac{z}{x}) dx = \int_{-\infty}^{+\infty} \frac{1}{|y|} f(\frac{z}{y}, y) dy (X,Y)∼f(x,y),Z=XY,fZ(z)=∫−∞+∞∣y∣f(yz,y)dy(X,Y) \sim f(x,y), Z=\frac{X}{Y}, f_Z(z) = \int_{-\infty}^{+\infty} |y| f(yz, y) dy