An immune system creates a burst of $N$ new white blood cells with probability $b$ per unit time. White blood cells die with probability $d$ each per unit time. Write down the master equation for $P_{n}(t)$, the probability that there are $n$ white blood cells at time $t$.

Given that $n=n_{0}$ initially, find an expression for the mean of $n$.

Show that the variance of $n$ has the form $A e^{-2 d t}+B e^{-d t}+C$ and find $A, B$ and $C$.

If the immune system were modified to produce $k$ times as many cells per burst but with probability per unit time divided by a factor $k$, how would the mean and variance of the number of cells change?