# The time to hatch between $k_1$ and $k_2$ eggs (if hatching times are independent and exponentially distributed)

I have $N$ eggs that each hatch after a time given by independent exponentially distributed random variables with identical rate parameters $\lambda \space seconds^{-1}$. At our initial time $t = 0$, none of the eggs have hatched.

As a function of the time $t$, what is the probability that (at a given time $t_i$) between $k_1$ and $k_2$ of the $N$ eggs have hatched (where $k_1 \leq k_2 \leq N$)? For another example, if we wait five seconds, we might want to ask: what is the probability P(at least two eggs have hatched AND at most five eggs have hatched)?

• Your question makes no sense and what have you done in class? – Lost1 Jan 12 '14 at 16:35
• @Lost1 What about my question makes no sense? I have some number of processes that occur at (originally unknown) times drawn from the same exponential distribution, and I want to know, as a function of time, the probability that a bounded number of them have occurred? – CowardlyLion Jan 12 '14 at 17:17
• Between $N \geq k_1$ and $N\geq k_2$ are hatched? – Lost1 Jan 12 '14 at 17:19
• @Lost1 Is this the confusing part? It might be written poorly. I just meant that between some integer number $k_1$ and $k_2$ of the $N$ eggs have hatched. Here $k_1$ and $k_2$ are both integers $\leq N$ and $k_1 \leq k_2$. For example, we could have $N = 10$ and $k_1 = 0$ and $k_2 = 5$. – CowardlyLion Jan 12 '14 at 17:21
• Hmm interesting... – Lost1 Jan 12 '14 at 17:24

## 1 Answer There is no nice way to write the sum, sorry.

• Ah, that makes sense. – CowardlyLion Jan 12 '14 at 17:39