Verified answer

f(x,y)=2 for 0>, y=0, x+y<1, 0 elsewhere

???

2011-07-17 18:57:59 補充：

if the joint probability density of X and Y is given by

f(x,y)=2 for x>0,y>0, x+y<1, 0 elsewhere

find (1)p(x≦1/2, y ≦1/2)p(x≦1/2, y ≦1/2)= [x=0 to 1/2] ∫ [y=0 to 1/2]∫2dydx= [x=0 to 1/2] ∫ [y=0 to 1/2]{2y}dx= [x=0 to 1/2] ∫1dx= [x=0 to 1/2]{x}=1/2(2)p(x+y>2/3)p(x+y>2/3)= [x=0 to 2/3] ∫ [y=2/3-x to 1-x]∫2dydx+ [x=2/3 to1] ∫ [y=0 to 1-x]∫2dydx= [x=0 to 2/3] ∫ [y=2/3-x to 1-x]{2y}dx+[x=2/3 to1] ∫ [y=0 to 1-x]{2y}dx= [x=0 to 2/3] ∫2/3*dx+[x=2/3 to1] ∫(2-2x)dx= [x=0 to 2/3]{2/3*x}+[x=2/3 to1](2x-x^2)=4/9+(2-1)-(4/3-4/9)=5/9

(3)p(x>2y)p(x>2y)= [x=0 to 2/3] ∫ [y=0 to x/2]∫2dydx+ [x=2/3 to1] ∫ [y=0 to 1-x]∫2dydx= [x=0 to 2/3] ∫ [y=0 to x/2]{2y}dx+[x=2/3 to1] ∫ [y=0 to 1-x]{2y}dx= [x=0 to 2/3] ∫xdx+[x=2/3 to1] ∫(2-2x)dx= [x=0 to 2/3]{x^2/2}+[x=2/3 to1](2x-x^2)=4/18+(2-1)-(4/3-4/9)=1/3

2011-07-17 19:01:34 補充：

第二題與第三題變數x與y的積分範圍，乃依變數限制式子不同而不同，請於平面坐標系作圖可求得，本人因點數不足無法貼圖，請見諒。