已知随机变量(X,Y)的概率密度为f(x,y)=Csin(x+y),0
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![已知随机变量(X,Y)的概率密度为f(x,y)=Csin(x+y),0](/uploads/image/z/11516658-42-8.jpg?t=%E5%B7%B2%E7%9F%A5%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%28X%2CY%29%E7%9A%84%E6%A6%82%E7%8E%87%E5%AF%86%E5%BA%A6%E4%B8%BAf%28x%2Cy%29%3DCsin%28x%2By%29%2C0)
已知随机变量(X,Y)的概率密度为f(x,y)=Csin(x+y),0
已知随机变量(X,Y)的概率密度为f(x,y)=Csin(x+y),0
已知随机变量(X,Y)的概率密度为f(x,y)=Csin(x+y),0
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] cosxsinyd...
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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
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