The lifetime of $N$ atoms is exponentially distributed with parameter $\lambda>0$. Suppose that the atoms decay independently of one another. I want to find the probability that exactly $k$ of the atoms decayed until time $T$. I also want to determine the time for which on average half of the atoms decay.
Let $0\leq t_1 \leq t_2 \leq \ldots \leq t_N$ be the lifetimes of the $N$ atoms. I think the probability I am looking for is
$$P(t_1, t_2, \ldots, t_k \leq T)=\prod_{i=1}^k P(t_i\leq T)= \prod_{i=1}^k \int_0^T \lambda e^{-\lambda t_i} dt_i .$$
Also, let $T_0>0$ be the time for which on average half of the $N$ atoms decay. To find $T_0$ do I let $k=N/2$ (maybe let it be the greatest integer less than $N/2$) and $T=T_0$ in the integral above?
Thanks!