Question

概率统计题目一道，求概率密度

设X，Y均服从均匀分布，X~U[0，2]，Y~U[0，1]，且X，Y独立，求Z=X+Y的概率密度

Answer 1

用二维概率密度公式来求
Fz(z)=∫∫[x+y<=z] f(x,y)dxdy=∫∫[x+y<=z] 1/2 dxdy
z<0时,Fz(z)=0,z>3时Fz(z)=1
z∈[0,1]时,Fz(z)=∫∫[x+y<=z] 1/2 dxdy=∫[0->z]∫[0->z-y] 1/2 dxdy=(z^2)/4
z∈[1,2]时,Fz(z)=∫∫[x+y<=z] 1/2 dxdy=∫[0->1]∫[0->z-y] 1/2 dxdy=z/2-1/4
z∈[2,3]时,Fz(z)=∫∫[x+y<=z] 1/2 dxdy
=∫[0->1]∫[0->z-y] 1/2 dxdy - ∫[2->z]∫[0->z-x] 1/2 dydx=3z/2-(z^2)/4-5/4
0, z<0或z>3
故f(x)= z/2, z∈[0,1]
1/2, z∈[1,2]
3/2-z/2, z∈[2,3]

Answer 2

Z~U[0,3]

追问

求过程！