Suppose my blog post gets spam comments at the rate of one per day, and that each spam comment gets spam comments in reply at the same rate. At the end of three days, how many spam comments are to be expected in total?
Solution
Let $n_t$ be the number of messages received by time $t$. Because the rate of change in the expectation for $n_t$ over time starts at $1$ and increases by exactly the number of messages already received:
\[\frac{d}{dt} E(n_t) = E(n_t) + 1\]The only real functions of $t$ that satisfy this are the constant function with the value $-1$ (which obviously won’t do) and every function of the form $Ce^t - 1$; and given that $E(n_0)$ is $0$, $C$ must be $1$. So:
\[E(n_t) = e^t - 1\]And in particular, $E(n_3)$ is $e^3-1$, or about $19$.