1. Random Variables

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]

2. Originally Posted by matty888
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 $\displaystyle \int_{x=0}^{2} \int_{y = 0}^1 xy f(x, y) \, dy \, dx$.

3. Thank you!!

Thanks a million mr. fantastic you are very helpful.
Could anyone confirm that my answers are right,not so sure myself
(a) K = 1/5
(b) 1.5466
(c) 3.741155
(d) 0.8177
Thanks

4. Originally Posted by matty888
Thanks a million mr. fantastic you are very helpful.
Could anyone confirm that my answers are right,not so sure myself
(a) K = 1/5
(b) 1.5466
(c) 3.741155
(d) 0.8177
Thanks
(a) Correct.
(b) Correct
(c) I get different. What did you get for E(X^2)? I get 2.5333 (correct to 4 decimal places).
(d) Correct.