Verified answer

if the joint probability density of x and y is given by f(x,y)= 24y(1-x-y) for x>0, y>0 , x+y<1 , 0 elsewhere

find

(a)the marginal density of x f(x)=[y=0 to 1-x]∫f(x,y)dy=[y=0 to 1-x]∫24y(1-x-y)dy=[y=0 to 1-x]{12y^2-12xy^2-8y^3}=12(1-x)^2-12x(1-x)^2-8(1-x)^3=4(1-x)^2[3-3x-2+2x]=4(1-x)^3，0<x<1

(b) the marginal density of y f(y)=[x=0 to 1-y]∫f(x,y)dx=[x=0 to 1-y]∫24y(1-x-y)dx=24y*[x=0 to 1-y]{x-x^2/2-xy}=24y*[1-y-(1-y)^2/2-(1-y)y]=24y(1-y)[1-(1-y)/2-y]=12y(1-y)^2，0<y<1

also determine whether the two random variables are independentf(x)*f(y)= 4(1-x)^3*12y(1-y)^2=48 y (1-x)^3* (1-y)^2≠f(x,y)x and y are not independent