You want to construct a uniform distribution in the two-dimensional space over a triangle.
Therefore, you consider the random variables X and Y, which are jointly uniformly distributed
over the triangle formed by the points (0,0), (1,0), and (0,1). Which of the following gives the
joint PDF of (X,Y)?
Hint: Compute the area of the triangle and consider how the support of the PDF must look
like.
(i) $f_{X,Y}(x, y) = \begin{cases} \frac{1}{2} & 0 \le y \le x \le 1\\ 0 & \text{Otherwise} \end{cases}$
(ii) $f_{X,Y}(x, y) = \begin{cases} 1 & 0 \le y \le x \le 1\\ 0 & \text{Otherwise} \end{cases}$
(iii) $f_{X,Y}(x, y) = \begin{cases} \frac{1}{2} & 0 \le x \le 1, 0 \le y \le 1\\ 0 & \text{Otherwise} \end{cases}$
(iv) $f_{X,Y}(x, y) = \begin{cases} 1 & 0 \le x \le 1, 0 \le y \le 1\\ 0 & \text{Otherwise} \end{cases}$
