Question

已知随机变量(X,Y)的概率密度为f(x,y)=Csin(x+y), 0<x,y<π/4,试确定常数C并求Y的边缘概率密度

Answer 1

f(x,y)=Csin(x+y)
∫∫[0,π/4] [0,π/4] f(x,y)dxdy
=∫∫[0,π/4] [0,π/4] Csin(x+y)dxdy
=∫∫[0,π/4] [0,π/4] C(sinxcosy+cosxsiny)dxdy
=C∫∫[0,π/4] [0,π/4] sinxcosydxdy+C∫∫[0,π/4] [0,π/4] cosxsinydxdy
=C∫∫[0,π/4]sinxdx [0,π/4] cosydy+C∫∫[0,π/4] cosxdx[0,π/4] sinydy
=C(-cosx)[0,π/4] siny[0,π/4]+Csinx[0,π/4](-cosy)[0,π/4]
=C(1-√2/2+√2/2+√2/2+1-√2/2)
=2C
=1
C=1/2
f(x,y)=1/2sin(x+y)
fy(x,y)=∫[0,π/4] 1/2sin(x+y)dx
=1/2∫[0,π/4] (sinxcosy+cosxsiny)dx
=1/2(-cosxcosy+sinxsiny)[0,π/4]
=1/2*(1-√2/2)cosy+√2/4siny

Answer 2