例1已知二维随机变量(X,Y)的分布函数为 F(x, y)=AB++ arcto 1求常数A,B,C。2求P0<X<2,0<Y<3} 解:F(a,∞)=AIB+"ⅢC+=1 F(-0,)=A|B、z IIC+ arct() F(x,00)=AB+arcto C 3π2 丌 →B=C=-A= 」P0<X52.0<Y≤3}=F0.0+F(2.3-F(03)-F(2例1.已知二维随机变量(X,Y)的分布函数为 ( , ) [ ( )][ ( )] y C arctg x F x y = A B + arctg + )] 3 )][ ( 2 F(x, y) A[B + arctg( C + arctg 1)求常数A，B，C。 2)求P{,} 0<X<2,0<Y<3} 解: ] 1 2 ][ 2 ( ∞ , ∞ ) = [ + + = π π F A B C 2 2 )] 0 3 ][ ( 2 ( −∞ , ) = [ − + = y F y A B C arctg π ] 0 2 )][ 2 ( , −∞ ) = [ + ( − = π C x F x A B arctg π 1 2 2 π ⇒ B = C = A = 1 P{0 < X ≤ 2 0 < Y ≤ 3} F(0 0) + F(2 3) F(0 3) F(2 0) 16 P{0 < X ≤ 2,0 < Y ≤ 3} = F(0,0) + F(2,3) − F(0,3) − F(2,0) =