theoretical biology - Modeling population growth - Variance

I've been studing population growth models, but there's something I can not find that's fustrating. That's a formula for the variance in population growth. I know that other models can be aplied, but I want to start with the simplest case.

Let's assume a population is growing exponentially as defined by: $N(t) = N(0)W^{t}$

The parameters $W$(absolute fitness in my definition) tell us that the time for division (if the species reproduces binarally) is given by an exponential distribution with $\lambda = \dfrac{\ln W}{\ln 2}$

So I see a model for $N(t)$ and a random variable associated with this model. What is the pdf, expected value and variance of $N(t)$? The expected value I expect it to be given by the first equation (or the value to be the same) but what about the variance?

Answer

Check out Kendall (1949), section 2. He shows that the pdf is a geometric distribution. In your notation, it's $P_n(t) = N(0)W^{-t}\left(1-W^{-t}\right)^{n-1}$, which implies that the mean is indeed $E[N(t)] = N(0)W^t$ and the variance is $\text{Var}[N(t)] = N(0) W^t(W^t-1)$. (Be careful -- his definition of $\lambda$ differs from yours by a factor of $\ln 2$.)