Algebra

Problem 1071

Sir Sagar0

Given p=\begin{bmatrix} 0.5 &0.5 \\ 0.9&0.1 \end{bmatrix}

and the initial-state matrix So = \begin{bmatrix} 1 &0 \\ \end{bmatrix}

Find S1 for the given initial-state matrix So and explain what it represents.

 

Solution:-

 

Recall that S1 is the proportion of the population that is in each state after 1 trial. To calculate S1, multiply the initial state matrix, S0, by the transition matrix,  P.

S1 = S0P

= \begin{bmatrix} 1 &0 \end{bmatrix}\cdot \begin{bmatrix} 0.5 &0.5 \\ 0.9&0.1 \end{bmatrix}

Now use matrix multiplication to calculate S1.

S1 = \begin{bmatrix} 1 &0 \end{bmatrix}\cdot \begin{bmatrix} 0.5 &0.5 \\ 0.9&0.1 \end{bmatrix}

= \begin{bmatrix} (1\cdot 0.5)+(0\cdot 0.9)) &(1\cdot 0.5)+(0\cdot 0.1) \end{bmatrix}

Simplify each entry in the first state matrix to finish calculating S1.

S1 = \begin{bmatrix} 1 &0 \end{bmatrix}\cdot \begin{bmatrix} 0.5 &0.5 \\ 0.9&0.1 \end{bmatrix}

= \begin{bmatrix} (1\cdot 0.5)+(0\cdot 0.9)) &(1\cdot 0.5)+(0\cdot 0.1) \end{bmatrix}

= \begin{bmatrix} 0.5 &0.5 \end{bmatrix}

Therefore S1 is \begin{bmatrix} 0.5 &0.5 \end{bmatrix}.

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