Random Variables

**matty888** · January 6th 2009, 08:54 AM

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]

**mr fantastic** · January 6th 2009, 09:12 AM

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 $\int_{x=0}^{2} \int_{y = 0}^1 f(x, y) \, dy \, dx = 1$ for K.

(b) Calculate $\int_{x=0}^{2} x \left( \int_{y = 0}^1 f(x, y) \, dy\right) \, dx$ .

(c) Calculate $E(X^2) = \int_{x=0}^{2} x^2 \left( \int_{y = 0}^1 f(x, y) \, dy\right) \, dx$ . Then $Var(X) = E(X^2) - [E(X)]^2$ .

(d) Calculate $\int_{x=0}^{2} \int_{y = 0}^1 xy f(x, y) \, dy \, dx$ .

**matty888** · January 6th 2009, 10:00 AM

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

**mr fantastic** · January 7th 2009, 04:42 AM

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.

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