Question

The random variables X and Y have a density of
fXY (x, y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, and fXY (x, y) = 0 elsewhere. Evaluate
E [X] and E [XY ].

Answer 1

Answer for Question #71378

$$f _ {X} (x) = \int_ {0} ^ {1} (x + y) d y = x + \frac {1}{2}$$ $$f _ {Y} (y) = \int_ {0} ^ {1} (x + y) d x = y + \frac {1}{2}$$ $$E [ X ] = \int_ {0} ^ {1} x \left(x + \frac {1}{2}\right) d x = \frac {7}{12}$$ $$E [ Y ] = \int_ {0} ^ {1} y \left(y + \frac {1}{2}\right) d y = \frac {7}{12}$$ $$E \left[ X Y \right] = E \left[ X \right] E \left[ Y \right] - K _ {X Y}, \quad K _ {X Y} = \int_ {0} ^ {1} \int_ {0} ^ {1} x y (x + y) d y d x = \frac {1}{3}$$ $$E [ X Y ] = \left(\frac {7}{12}\right) ^ {2} - \frac {1}{3} = \frac {49}{144} - \frac {48}{144} = \frac {1}{144}$$

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