Consider a pair of random variables X; Y with constant joint density on the quadrilateral with vertices (0; 0), (2; 0), (2; 6), (0; 12). a) Find the expected value E(X). b) Find the expected value E(Y ).

The given quadrilateral (call it Q) is a trapezoid with "base" lengths of 6 and 12, and "height" 2, so its area is (6 + 12)/2*2 = 18. This means the joint density is

where Q is the set of points

(y = 12 - 3x is the equation of the line through the points (0, 12) and (2, 6))

Recall the definition of expectation:

(a) Using the definition above, we have

(b) Likewise,

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