# Problem 1: Death Process¶

In this problem, we study a death process, which has the unique reaction $A \to \emptyset$ with rate constant $\delta$.

**a)** Write down an ODE describing the deterministic kinetics.

**b)** Write down the Chemical Master Equation for the stochastic kinetics. Qualitatively compare the two.

**c)** Find the expected value of the extinction time $T_0$ when starting from an initial population of $n$.

**d)** Find the variance of the extinction time $T_0$.

**e)** Consider two independent death processes, one for species $A$ starting at an initial population of $n_A$ and one for species $B$ starting at an initial population of $n_B$. Denoting by $T_A$ the extinction time of $A$ and by $T_B$ that of $B$, find (an estimation of) the probability that $T_A \leq T_B$.

# Problem 2: Birth and Death Process¶

In this problem, we add a birth reaction to the death process from the previous problem, i.e., the reaction $A \to A + A$ with rate constant $\gamma$.

**a)** Write down an ODE describing the deterministic kinetics.

**b)** Write down the Chemical Master Equation for the stochastic kinetics. Qualitatively compare the two.

**c)** Find the probability of extinction when starting from an initial population of $n$.

**d)** For the case of almost-sure extinction, find the expected extinction time. Evaluate this formula for $n=1,2,3$.