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Key Equations

(14-1)

π (i) = (π_{1}, π_{2}, π_{3}, \dots,, π_{n})

$π (i) = (π_{1}, π_{2}, π_{3}, \dots,, π_{n})$

Vector of state probabilities for period i.

(14-2)

P = [\begin{matrix} P_{11} & P_{12} & P_{13} & \dots & P_{1 n} \\ P_{21} & P_{22} & P_{23} & \dots & P_{2 n} \\ ⋮ & ⋮ \\ P_{m 1} & P_{m n} \end{matrix}]

$P = [\begin{matrix} P_{11} & P_{12} & P_{13} & \dots & P_{1 n} \\ P_{21} & P_{22} & P_{23} & \dots & P_{2 n} \\ ⋮ & ⋮ \\ P_{m 1} & P_{m n} \end{matrix}]$

Matrix of transition probabilities—that is, the probability of going from one state into another.

(14-3)

π (1) = π (0) P

$π (1) = π (0) P$

Formula for calculating the state 1 probabilities given state 0 data.

(14-4)

π (n + 1) = π (n) P

$π (n + 1) = π (n) P$

Formula for calculating the state probabilities for the period $n + 1$ $n + 1$ if we are in period n.

(14-5)

π (n) = π (0) P^{n}

$π (n) = π (0) P^{n}$

Formula for computing the state probabilities for period n if we are in period 0.

(14-6)

π = π P

$π = π P$

Equilibrium state equation used to derive equilibrium probabilities.

(14-7)

P = [\begin{matrix} I & O \\ A & B \end{matrix}]

$P = [\begin{matrix} I & O \\ A & B \end{matrix}]$

Partition of the matrix of transition for absorbing state analysis.

(14-8)

F = {(I - B)}^{- 1}

$F = {(I - B)}^{- 1}$

Fundamental matrix used in computing probabilities of ending up in an absorbing state.

(14-9)

[\begin{matrix} a & b \\ c & d \end{matrix}]^{- 1} = [\begin{matrix} \frac{d}{r} & \frac{- b}{r} \\ \frac{- c}{r} & \frac{a}{r} \end{matrix}] where r = a d - b c

$[\begin{matrix} a & b \\ c & d \end{matrix}]^{- 1} = [\begin{matrix} \frac{d}{r} & \frac{- b}{r} \\ \frac{- c}{r} & \frac{a}{r} \end{matrix}] where r = a d - b c$

Inverse of a matrix with 2 rows and 2 columns.

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Table of Contents for Key Equations

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Table of Contents for
Key Equations