Let (X, Y ) be a random point chosen uniformly on the region R = {(x, y) : |x| + |y| ≤ 1} b. Find the marginal densities of X and Y.

Question

I have graphed the set and realised that x is in [-1,1] and so is y. However I cannot understand what am I to integrate?

Answer 1

Let $\alpha$ be the area of $\mathrm{R}.$ Then $f_{(X,Y)}(x,y)= \dfrac{1}{\alpha} \mathbf{1}_{\mathrm{R}}(x,y).$ Let $\mathrm{R}_1(y)$ be the section in the $x$-axis corresponding to $y;$ that is to say, $\mathrm{R}_1(y) = \{x \mid (x,y) \in \mathrm{R}\}.$ Define $\mathrm{R}_2(x)$ mutatis mutandis. Denote by $\alpha_1(y)$ and $\alpha_2(x)$ the corresponding lengths of $\mathrm{R}_1(y)$ and $\mathrm{R}_2(x).$ Then, whenever $\alpha_1(y) > 0,$ we have $f_{X \mid Y = y}(x) = \dfrac{1}{\alpha_2(y)}\mathbf{1}_{\mathrm{R}_1(y)}(x),$ similar expression for $f_{Y\mid X = x}(y).$ It is up to you to find the values of the $\alpha$'s.