Problem

乙、計算、證明題：共 3 大題，每大題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。

3. Consider the function

$$f(x,y)= \begin{cases} ke^{-(x+y)/a}, & \text{if }x\ge 0,\ y\ge 0,\\ 0, & \text{elsewhere.} \end{cases}$$

Find the relationship between the positive constants $a$ and $k$ such that $f$ is a joint probability density function of the continuous random variables $x$ and $y$.