Quantitative Analysis for Management, 13/e

14.2 Matrix of Transition Probabilities

The concept that allows us to get from a current state, such as market shares, to a future state is the matrix of transition probabilities. This is a matrix of conditional probabilities of being in a future state given a current state. The following definition is helpful:

$$\text{Let} P_{ij}=\text{conditional} \text{probability} \text{of} \text{being} \text{in} \text{state} j \text{in} \text{the} \text{future} \text{given} \text{the} \text{current} \text{state} \text{of} i$$

The matrix of transition probabilities allows us to get from a current state to a future state.

For example, $P_{12}$ is the probability of being in state 2 in the future given the event was in state 1 in the period before:

Let $P=$matrix of transition probabilities

$$P=\left[\begin{array}{lllll}{P}_{11}\hfill & P_{12}\hfill & P_{13}\hfill & \dots \hfill & P_{1n}\hfill \\ P_{21}\hfill & P_{22}\hfill & P_{23}\hfill & \dots \hfill & P_{2n}\hfill \\ ⋮\hfill & ⋮\hfill \\ P_{m1}\hfill & P_{mn}\hfill \end{array}\right]$$ (14-2)

Individual $P_{ij}$ values are usually ...