Question

Let \(\{X(t)\}_{t\geq 0}\) be a linear pure death process with death rate \(\mu_i = 5i\), \(i = 0, 1, \dots, N\), \(N \geq 1\). Suppose that \(p_i(t) = P(X(t) = i)\). Then the system of forward Kolmogorov’s equations is

• A. \(\frac{dp_i(t)}{dt} = -\mu_i p_i(t) + \mu_{i+1} p_{i+1}(t)\)
• B. \(\frac{dp_i(t)}{dt} = -\mu_i p_i(t) + \mu_{i-1} p_{i-1}(t)\)
• C. \(\frac{dp_i(t)}{dt} = -\mu_i p_i(t) + \mu_{i+1} p_{i+1}(t) + \mu_{i-1} p_{i-1}(t)\)
• D. \(\frac{dp_i(t)}{dt} = -\mu_i p_i(t) + \mu_{i-1} p_{i-1}(t)\)

Hint:

In pure death processes, the forward Kolmogorov equations relate the rate of change of the probabilities to the death rates and the transition probabilities from neighboring states.

Answer 1

The Correct Option is B

In a linear pure death process, the rate of change of the probability \(p_i(t)\) of being in state \(i\) is given by the difference between the rate of death from state \(i\) and the rate of transition to state \(i\) from the previous state \(i-1\). The Kolmogorov forward equations for such a process are: \[ \frac{dp_i(t)}{dt} = -\mu_i p_i(t) + \mu_{i-1} p_{i-1}(t) \] This equation represents the probability of being in state \(i\), considering both the death rate and the transition probabilities from the previous state \(i-1\). Thus, the correct answer is option (B).