## $T_1, … , T_n$ time of life of instruments with geometric distribution of parameter p independent

$$T_1, … , T_n$$ time of life of instruments with geometric distribution of parameter p independents. I define S as the first n instants of failure and U as the last n instant of failure. I want to find the law of S and the density of U.

This is what I did so far:

To find the law of S I start by calculating: $$\mathbb{P}(S = k) = \mathbb{P}(S \geq k) – \mathbb{P}(S \geq k + 1)$$ $$\mathbb{P}(S = k) = \mathbb{P}(S < k+1) – \mathbb{P}(S < k)$$ But here I am stuck on what I should do because I still do not know the distribution.

I do the same reasoning with U as follows: $$\mathbb{U}(S = k) = \mathbb{U}(U \leq k) – \mathbb{U}(S \leq k + 1)$$ But I still get nowhere.

Any suggestions?