A pair of random variables X and Y have continuous joint density function f(x,y) on (0,2)x(0,1) given by
f(x,y) = K(x^3 + xy) 0<x<2 , 0<y<1.
(a) Determine K
(b) Determine E[X]
(c) Determine V[X]
(d) Determine E[XY]
(a) Solve $\displaystyle \int_{x=0}^{2} \int_{y = 0}^1 f(x, y) \, dy \, dx = 1$ for K.
(b) Calculate $\displaystyle \int_{x=0}^{2} x \left( \int_{y = 0}^1 f(x, y) \, dy\right) \, dx$.
(c) Calculate $\displaystyle E(X^2) = \int_{x=0}^{2} x^2 \left( \int_{y = 0}^1 f(x, y) \, dy\right) \, dx$. Then $\displaystyle Var(X) = E(X^2) - [E(X)]^2$.
(d) Calculate $\displaystyle \int_{x=0}^{2} \int_{y = 0}^1 xy f(x, y) \, dy \, dx$.