Question 10.6.3: When the Atlanta Braves data from Example 2 is used to const...
When the Atlanta Braves data from Example 2 is used to construct the transition matrix (not shown here), it turns out that the sum of the first column of the fundamental matrix M is 4.5048, and the first column of the matrix SM is
\left[\begin{array}{l}.3520 \\.3309 \\.2365 \\.0805\end{array}\right].
Compute the number of earned runs the Braves can expect to score per inning based on their performance in 2002. How many earned runs does the model predict for the entire season, if the Braves play 1443 \frac{2}{3} innings, as they did in 2002?
The first column of SM shows that, for example, the probability that the Braves left no runners on base is .3520. The expected number of runners left on base is
E[L]=0(.3520)+1(.3309)+2(.2365)+3(.0805)=1.0454.
The expected number of batters is E[B] = 4.5048, the sum of the first column of M: From Equation (6), the expected number of earned runs E[R] is
E[R]=E[B]-E[L]-3 (6).
E[R]=E[B]-E[L]-3=4.5048-1.0454-3=.4594.
The Markov chain model predicts that the Braves should average .4594 earned run per inning. In 1443 \frac{2}{3} innings, the total number of earned runs expected is
.4594 \times 1443.67=663.22.
The actual number of earned runs for the Braves in 2002 was 636, so the model’s error is 27.22 runs, or about 4.3%.