Since joint pdf is not given in the question hence we can assume that
Joint pdf of variables X and Y is
fX,Y(x,y)=6xy when 0≤x≤1,0≤y≤x
Otherwise
a.) fX(x)=∫−∞∞fXY(x,y)dy
=∫0x6xydy
=3x2
Thus, fX(x)=3x20≤x≤1
0 Otherwise
To find fY(y)for0≤y≤1 , we can write,
fY(y)=∫−∞∞fXY(x,y)dx
=∫y216xydx
=3y(1−y4)
fY(y)=3y(1−y4) 0≤y≤1
=0 Otherwise
b.) The conditional pdf,
fY∣X(y∣x)=fX(x)fX,Y(x,y)
=3x26xy=x2y
fX∣Y(x∣y)=fY(x)fX,Y(x,y)
=3y(1−y4)6xy=(1−y4)2x
c.) The E(X/Y=y)
=∫−∞∞xfX∣Y(x∣y)dx
=∫y21x1−y42xdx
=3(1−y4)2(1−y6)
d.) Yes X and Y are statistically independent.
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