Joint distribution of min(X,Z) and min(Y,Z)

Question

I have three random variables, $X\sim exp(a), Y\sim exp(b), Z \sim exp(c).$ They are all independent. I need to find $P(Min(X,Z)\geq u_1, Min(Y,Z)\geq u_2)$ I.e the joint probability of the two minimum.

I think that I somehow need to break the minimum and have three inqualities. After that Ican use their independece. The problem is that I don't know how to break them up.

Any help would be much helpful.

Answer 1

Hint:

On base of rule: $$\min(a,b)\geq c\iff a\geq c\text{ and }b\geq c$$ you can find that the event can also be described as: $$\{X\geq u_1,Y\geq u_2,Z\geq\max(u_1,u_2)\}$$Now apply independence.