Calculating growth of an exponential distribution

Question

A bacterium divides in to two bacteria at time zero. The time until bacteria divides further is an exponential random variable, with $\lambda=1$. What is the expected time $T_n$ when bacteria reach the number $n$?

I tried thinking along the relation between Poisson and exponential distribution but it got me even more confused. I am working on this approach now.

Assuming there are $r$ bacteria at the moment, and calculating the expected duration to reach the population of $(r+1)$ bacteria denoted by $T_r$.

$T_r=\int\limits_{0}^{\infty} rte^{-t}(1-e^{-t})^{r-1}dt$.

Therefore the expected duration for the bacteria to reach $n$ is $\sum\limits_{r=2}^{n-1}T_r$.

How can I do this?

Answer 1

Consider the sequence $T_3-T_2, T_4-T_3,\ldots$. Clearly $\mathbb P(T_1=0)=\mathbb P(T_2=0)=1$. Now for each $k\geqslant 2$, we have $T_{k+1}-T_k\sim\mathrm{Expo}(k\lambda)$ as the time until a bacteria division is the minimum of $k$ $\mathrm{Expo}(\lambda)$ random variables. Therefore $\mathbb E[T_{k+1}-T_k] = \frac1{k\lambda}$. Write $$ T_n = \sum_{k=2}^{n-1} (T_{k+1}-T_k) , then $$ \begin{align} \mathbb E[T_n] &= \mathbb E\left[\sum_{k=2}^{n-1} (T_{k+1}-T_k)\right]\\ &= \sum_{k=2}^{n-1}\mathbb E[T_{k+1}-T_k]\\ &= \sum_{k=2}^{n-1} \frac1{k\lambda}\\ &=\frac1\lambda \left(-1+\sum_{i=1}^{n-1}i\right). \end{align}

Answer 2

An exponential r.v. has a memoryless property.

At time $t = 0$, there are two bacteria. At time $t = T_1$, there are $2^2$ bacteria. At time $t= T_1 + T_2$, there are $2^3$ bacteria and so on ...

here $T_i$ represents the time BETWEEN the $(i-1)$th division and $i$th division; each such $T_i \sim Exp(\lambda)$. In particular, $T_1$ is the time until the first division after zero.

Let $k$ be the smallest integer such that $2^{k+1} > n$. Then, $T = T_1 + T_2 + \cdots + T_k$ is the time it takes for the bacteria population to exceed or become equal to $n$.

so

$\begin{align*}\mathbb{E}[T] &= \mathbb{E} [T_1 + \cdots + T_k]\\ &= k \mathbb{E}[T_1]\\ &= \frac{k}{\lambda} \end{align*}$