Question

设随机变量概率密度函数f(x)=Ce^(-λ|x|),Y=|X|,求(X,Y)的联合分布函数

Answer 1

首先求C,∫ce^(-s|x|)dx |-无穷大，无穷大=2∫ce^(-sx)dx|0,正无穷大
=-2c/s e^(-sx)|0,正无穷大=2c/s =1 => c=s/2, s就是lamda
当y <=0时，P(X<x, Y<y)=0
当x>y>0时，P(X<x, Y<y) = P(X<x, |X|<y) =P(-y<X<y) = s/2∫e^(-s|x|)dx |-y,y
=s∫e^(-sx)dx |0,y = -e^(-sx)|0,y = 1-e^(-sy)
当0<x<y时

P(X<x, Y<y) = P(X<x, |X|<y) =P(-y<X<x) = s/2∫e^(-s|t|)dt |-y,x =s/2 ∫e^(st)dt |-y,0 +s/2 ∫e^(-st)dt |0,x= -1/2 e^(st)|-y,0 - 1/2 e^(-st)|0,x = -1/2 +1/2e^(-sy) +1/2 e^(-sx)
当-y<x<0<y时，P(X<x, Y<y) = P(X<x, |X|<y) =P(-y<X<x) = s/2 ∫e^(st)dt |-y,x = -1/2 e^(st)|-y,x = -1/2 e^(sx) +1/2 e^(-sy)
当x<-y<0时，P(X<x, Y<y) = P(X<x,|X|<y) =0