Используем свойство двумерной плотности вероятности:
[b] ∫ ∫ _((D))f(x;y)dxdy=1[/b]
∫ ^(+∞)_(-∞) ∫ ^(+∞)_(-∞)Cdxdy/(4+x^2)(9+y^2)=1;
C*(1/2)*arctg(x/2)|^(+∞)_(-∞)*(1/3)*arctg(y/3)|^(+∞)_(-∞)=1;
C*(1/6)*((Pi/2)-(-Pi/2))*((Pi/2)-(-Pi/2))=1
C*(1/6)*(Pi)^2=1 ⇒ [b] C=6/(Pi)^2 [/b]
б)
F(x;y) = ∫ ^(y)_(-∞) ∫ ^(x)_(-∞)f(x;y)dxdy =
= (6/(Pi)^2)*∫ ^(y)_(-∞) ∫ ^(x)_(-∞)dxdy/(4+x^2)(9+y^2) =
=(6/(Pi)^2)*(1/2)*arctg(x/2)|^(x)_(-∞)*(1/3)*arctg(y/3)|^(y)_(-∞)=
=(1/2Pi)*(artg(y/2)-(-Pi/2))*(arctg(x/2)-(-Pi/2))
=(1/(Pi)^2)*(artg(y/2)+(Pi/2))*(arctg(x/2)+(Pi/2))
=[b]((1/Pi)arctg(x/2)+(1/2))* ((1/Pi)arctg(y/3)+(1/2))[/b]
О т в е т.
a) 6/(Pi)^2;
б) F(x;y) =((1/Pi)arctg(x/2)+(1/2))* ((1/Pi)arctg(y/3)+(1/2))