> 
> A random variable $(T\_1, T\_2)$ is called two-dimensional exponentially
>  distributed with parameters $(\lambda\_1, \lambda\_2, \lambda\_3)$, if
>  the distribution function has the form
> 
> 
> $$F(t\_1,t\_2) = \left\{\begin{matrix}
> 1-e^{-(\lambda\_1+\lambda\_3)t\_1}-e^{-(\lambda\_2+\lambda\_3)t\_2}+e^{-\lambda\_1t\_1-\lambda\_2t\_2-\lambda\_3 \text{max}(t\_1,t\_2)}, \text{ if } t\_1>0, t\_2>0\\ 0
>  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{
> else } \end{matrix}\right.$$
> 
> 
> Show that: $T\_1$ and $T\_2$ are independent and exponentially distributed, if and only if $\lambda\_3 = 0$
> 
> 
> 

This is from exam of last year and I like to know how to do this correct. From my other question I'm very happy I almost do it correct and now know better how it work: [Determine the marginal distributions of $(T\_1, T\_2)$](https://math.stackexchange.com/questions/2604299/determine-the-marginal-distributions-of-t-1-t-2)

So we already know $$\lim\_{t\_1 \rightarrow \infty}f(t\_1,t\_2) = 1-e^{-\infty}-e^{-(\lambda\_2+\lambda\_3)t\_2}+e^{-\infty} = 1+0-e^{-(\lambda\_2+\lambda\_3)t\_2}+0=1-e^{-(\lambda\_2+\lambda\_3)t\_2}$$

$$\lim\_{t\_2 \rightarrow \infty}f(t\_1,t\_2) = 1-e^{-(\lambda\_1+\lambda\_3)t\_1}-e^{-\infty}+e^{-\infty} = 1-e^{-(\lambda\_1+\lambda\_3)t\_1}-0+0 = 1-e^{-(\lambda\_1+\lambda\_3)t\_1}$$

But I think this is not really useful for answer the question.. I try insert the given $\lambda\_3=0$ and see maybe we can form it then:

$$F(t\_1,t\_2) = \left\{\begin{matrix}
1-e^{-\lambda\_1t\_1}-e^{-\lambda\_2t\_2}+e^{-\lambda\_1t\_1-\lambda\_2t\_2 }, \text{ if } t\_1>0, t\_2>0\\ 0
 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{
else } \end{matrix}\right.$$

$$\Leftrightarrow F(t\_1,t\_2) = \left\{\begin{matrix}
1-e^{-\lambda\_1t\_1}-e^{-\lambda\_2t\_2}+\frac{e^{-\lambda\_1t\_1}}{e^{\lambda\_2t\_2}}, \text{ if } t\_1>0, t\_2>0\\ 0
 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{
else } \end{matrix}\right.$$

$$\Leftrightarrow F(t\_1,t\_2) = \left\{\begin{matrix}
1-e^{\lambda\_2t\_2} \left(\frac{e^{-\lambda\_1t\_1}}{e^{\lambda\_2t\_2}}+1-e^{-\lambda\_1t\_1}\right), \text{ if } t\_1>0, t\_2>0\\ 0
 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{
else } \end{matrix}\right.$$