Question

g describe the continuous time process with generator matrix that is, the exponential times in each state and probabilities of going from and to the different states.

Answer 1

The continuous time process with generator matrix describes the exponential times in each state and the probabilities of transitioning between different states.

Step-by-step explanation:

The continuous time process with generator matrix describes the exponential times in each state and the probabilities of transitioning between different states. The generator matrix, denoted as G, is a square matrix where each entry represents the rate of transition from one state to another. The exponential times in each state describe the time spent in that state before transitioning to another state.

To calculate the probabilities of transitioning between states, we use the generator matrix. Let's say we have a 3-state system with generator matrix G:

G = [[-λ₁, λ₁, 0], [λ₂, -λ₂-λ₃, λ₃], [0, λ₄, -λ₄]]

The negative values on the diagonal represent the rates of leaving each state, while the positive values off the diagonal represent the rates of transitioning to other states. The probabilities can be calculated using the formula: P(t) = e^(Gt), where P(t) is the transition probability matrix at time t.