# Bacteria that doubles every time step with a probability of p or dies. What is expected number of bacteria after n time steps?

As stated in the problem. Given a bacterium that at every time step $$t$$, either divides with a probability $$p$$ or dies off. What is the expected number of bacteria after $$n$$ time steps?

Does this use a binomial theorem? Markov Chains? I have no clue..

Experimental results show that with $$p = 0.75$$, here are the numbers per timestep $$t$$ (with $$10^7$$ simulations):

 | t | population | |----|------------| | 1 | 1.500407 | | 2 | 2.2507928 | | 3 | 3.376627 | | 4 | 5.0640936 | | 5 | 7.5959494 | | 6 | 11.3945926 | | 7 | 17.0909234 | | 8 | 25.6368554 | | 9 | 38.4575074 | | 10 | 57.6884722 | | 11 | 86.5342376 | | 12 | 129.802634 | 

So the experimental results have shown that this follows a trend of $$(2p)^i$$ where $$i$$ is the timestep. I am not sure why that is though... any ideas?

• It seems like with $p=0.25$, the population should eventually reach $0$: the bacterium is more likely to die off than to split. (In fact, $\frac34$ of the time, the bacterium dies without ever splitting.) Or am I misunderstanding the setup? – Misha Lavrov Oct 4 '18 at 2:29
• Oh sorry, it's the opposite. It's 0.75! Thanks for that :) – QuantumHoneybees Oct 4 '18 at 2:30

If there are $$n$$ bacteria at time $$t$$, then on average $$pn$$ of them split and $$(1-p)n$$ die, giving an expected number of $$2pn$$ bacteria at time $$t+1$$.
If we start with one bacterium at time $$t=0$$, then by induction at time $$t$$ the expected number of bacteria is $$(2p)^t$$. For example, when $$t=10$$ and $$p=0.75$$, this gives us about $$57.665$$ bacteria in expectation.
More generally, the distribution of the bacteria follows a branching process. In particular, we can compute the distribution for any $$t$$ with generating functions: let $$f(x) = 2px^2 + (1-p)$$. Then the probability that there are $$n$$ bacteria at time $$t$$ is the coefficient of $$x^n$$ in the $$t$$-fold composition $$\underbrace{f(f(f(\cdots(f(}_tx))\cdots))).$$ But deriving further properties of this distribution takes some work.
• So I did, I actually have the wikipedia page open on another tab. But that seems to deal with populations where the recurrence relation is described as $Z_{{n+1}}=\sum _{{i=1}}^{{Z_{n}}}X_{{n,i}}$, isn't the relation more of a product and not a sum? – QuantumHoneybees Oct 4 '18 at 2:40
• You can think of each bacterium as having either $0$ or $2$ children. In that case, the next generation's population is a sum of the number of children of all the bacteria in the current generation, which is exactly the branching process setup. – Misha Lavrov Oct 4 '18 at 2:42