finding the probability that exactly $k$ of $N$ atoms decay until time $T$ is the lifetime of the atoms is exponentially distributed

Question

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!

Answer 1

With the integral you might run into problems as there are $N \choose k$ possibilities for exactly $k$ out of $N$ random variables to fall below your bound $T$. It think it will be easier to connect your problem to a binomial distribution. Define an indicator $Y_i$ as follows \begin{align} Y_i = \begin{cases} 1, X_i \leq T\\ 0, X_i > T \end{cases} \end{align} Now with $p = \Pr[X_i \leq T]$ you can consider the sum $S = \sum_{i=1}^N Y_i$ which will follow a binomial distribution with parameters $p$ and $N$. Then you can easily answer questions like $\Pr[S = k]$. To find your $T_0$ now consider the quantile of the binomial distribution at $N/2$ as a function of $T$ (via $p$).