随机变量(X，Y)的概率密度
f(x,y)=
{A[R-√(x²+y²)],x²+y²≤R²;
0,其他
求：(1)系数A；
(2)(X，Y)落入圆x²+y²=r²2(r﹤R)内的概率．

随机变量(X，Y)的概率密度
f(x,y)=
{A[R-√(x²+y²)],x²+y²≤R²;
0,其他
求：(1)系数A；
(2)(X，Y)落入圆x²+y²=r²2(r﹤R)内的概率．

(1)∫^+∞_-∞∫^+∞_-∞f(x,y)dxdy =A∫∫_{x²+y²≤R²}[R-√(x²+y²)]dxdy =A∫∫_p≤R(R-p)pdpdθ =A∫^2π₀dθ∫^R₀(Rp-p²)dp =A(πR³/3)=1， 故A=3/πR³． (2)设D为圆域x²+y²﹤r²，则P((X，Y)∈D)=∫∫(3/πR³)(R-p)pdpdθ =3/πR³∫^2π₀dθ∫^x₀(Rp-p²)dp=(3r²/R²)(1-2r/3R)