设X,Y的联合概率密度为f(x,y)=1,0＜x＜1,|y|＜x,其他是0 求D(X),D(Y).

Question

设X,Y的联合概率密度为f(x,y)=1,0＜x＜1,|y|＜x,其他是0 求D(X),D(Y).

Answer 1

E(x)=∫[0,1]xdx∫[-x,x]dy=∫[0,1]2x^2dx=2/3
E(y)=∫[0,1]dx∫[-x,x]ydy=0
E(x^2)=∫[0,1]x^2dx∫[-x,x]dy=∫[0,1]2x^3 dx=1/2
E(y^2)=∫[0,1]dx∫[-x,x]y^2dy=∫[0,1]2x^3/3dx=1/6
D(x)=E(x^2)-[E(x)]^2)=1/18
D(y)=1/6