Let A ⊂ R2 be the triangle with vertices (0, 0), (1, 0) and (0, 1). They (X, Y ) are uniformly distributed in A. (a) Determine the probability density of (X,Y ). (b) Show that X and Y are dependent. Provide both an intuitive reasoning and a formal proof. (c) The distribution function of X for 0 ≤ x ≤ 1 is given by FX(x) = 2x −x2 Show this in two ways: (i) using the probability density and (ii) by means of a direct area calculation.

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