7.1.8 A baseball player is to play in the World Series.

Based upon his season play, you estimate that if he comes to bat four times in a game the number of hits he will get has a distribution pX =

0 1 2 3 4
.4 .2 .2 .1 .1

Assume that the player comes to bat four times in each game of the series.

(a)

Let X denote the number of hits that he gets in a series. Using the program NFoldConvolution, find the distribution of X for each of the possible series lengths: four-game, five-game, six-game, seven-game.

The convolve function in r will be useful in calculating the distribution throughout the series. We need to set type to open for a linear convolution of different lengths.

p_x <- c(0.4,0.2,0.2,0.1,0.1)
p_2 <- convolve(p_x,p_x, type = "open")
p_3 <- convolve(p_2, p_x, type = "open")
p_4 <- convolve(p_3, p_x, type = "open")
p4_mt <- t(matrix(c(c(0:16), p_4), ncol = 2))
p4_mt
##       [,1]   [,2]   [,3]   [,4]   [,5]   [,6]  [,7]   [,8]   [,9]  [,10]
## [1,] 0e+00 1.0000 2.0000 3.0000 4.0000 5.0000 6.000 7.0000 8.0000 9.0000
## [2,] 4e-04 0.0014 0.0044 0.0101 0.0228 0.0398 0.066 0.0929 0.1252 0.1378
##        [,11]   [,12]   [,13]   [,14]   [,15]  [,16]   [,17]
## [1,] 10.0000 11.0000 12.0000 13.0000 14.0000 15.000 16.0000
## [2,]  0.1452  0.1276  0.1056  0.0616  0.0368  0.016  0.0064
barplot(p4_mt[2,], names.arg = p4_mt[1,])

p_5 <- convolve(p_4, p_x, type = "open")
p5_mt <- t(matrix(c(c(0:20), p_5), ncol = 2))
p5_mt
##       [,1]    [,2]    [,3]    [,4]    [,5]    [,6]    [,7]    [,8]    [,9]
## [1,] 0e+00 1.00000 2.00000 3.00000 4.00000 5.00000 6.00000 7.00000 8.00000
## [2,] 4e-05 0.00018 0.00066 0.00181 0.00461 0.00972 0.01892 0.03245 0.05209
##      [,10]    [,11]    [,12] [,13]   [,14]    [,15]    [,16]    [,17]
## [1,] 9.000 10.00000 11.00000 12.00 13.0000 14.00000 15.00000 16.00000
## [2,] 0.074  0.09832  0.11704  0.13  0.1264  0.11456  0.08976  0.06416
##         [,18]   [,19]    [,20]    [,21]
## [1,] 17.00000 18.0000 19.00000 20.00000
## [2,]  0.03584  0.0192  0.00768  0.00256
barplot(p5_mt[2,], names.arg = p5_mt[1,])

p_6 <- convolve(p_5, p_x, type = "open")
p6_mt <- t(matrix(c(c(0:24), p_6), ncol = 2))
p6_mt
##       [,1]    [,2]    [,3]     [,4]     [,5]     [,6]     [,7]     [,8]
## [1,] 0e+00 1.0e+00 2.0e+00 3.000000 4.000000 5.000000 6.000000 7.000000
## [2,] 4e-06 2.2e-05 9.2e-05 0.000291 0.000826 0.001999 0.004412 0.008727
##          [,9]    [,10]     [,11]     [,12]     [,13]     [,14]     [,15]
## [1,] 8.000000 9.000000 10.000000 11.000000 12.000000 13.000000 14.000000
## [2,] 0.016026 0.026771  0.041708  0.059734  0.080004  0.098312  0.112832
##          [,16]     [,17]     [,18]     [,19]     [,20]     [,21]     [,22]
## [1,] 15.000000 16.000000 17.000000 18.000000 19.000000 20.000000 21.000000
## [2,]  0.118528  0.115584  0.101424  0.082112  0.058592  0.037696  0.019968
##          [,23]     [,24]     [,25]
## [1,] 22.000000 23.000000 24.000000
## [2,]  0.009728  0.003584  0.001024
barplot(p6_mt[2,], names.arg = p6_mt[1,])

p_7 <- convolve(p_6, p_x, type = "open")
p7_mt <- t(matrix(c(c(0:28), p_7), ncol = 2))
p7_mt
##       [,1]    [,2]     [,3]     [,4]      [,5]      [,6]      [,7]
## [1,] 0e+00 1.0e+00 2.00e+00 3.00e+00 4.0000000 5.0000000 6.0000000
## [2,] 4e-07 2.6e-06 1.22e-05 4.35e-05 0.0001361 0.0003679 0.0009013
##           [,8]      [,9]     [,10]      [,11]      [,12]    [,13]
## [1,] 7.0000000 8.0000000 9.0000000 10.0000000 11.0000000 12.00000
## [2,] 0.0019953 0.0040879 0.0077071  0.0135633  0.0221944  0.03408
##           [,14]      [,15]      [,16]      [,17]      [,18]      [,19]
## [1,] 13.0000000 14.0000000 15.0000000 16.0000000 17.0000000 18.0000000
## [2,]  0.0488284  0.0657452  0.0826928  0.0976416  0.1072976  0.1103088
##           [,20]      [,21]      [,22]     [,23]      [,24]      [,25]
## [1,] 19.0000000 20.0000000 21.0000000 22.000000 23.0000000 24.0000000
## [2,]  0.1048832  0.0925696  0.0744768  0.055072  0.0363008  0.0214784
##          [,26]      [,27]      [,28]     [,29]
## [1,] 25.000000 26.0000000 27.0000000 2.800e+01
## [2,]  0.010752  0.0048128  0.0016384 4.096e-04
barplot(p7_mt[2,], names.arg = p7_mt[1,])

(b)

Using one of the distribution found in part (a), find the probability that his batting average exceeds .400 in a four-game series. (The batting average is the number of hits divided by the number of times at bat.)

There are 20 at-bats for a 5 game series. He would need 8 or better hits to exceed a batting average of 0.400 for the series. We can simply sum the P_5 series above from 8 to 20 to get the odds of this batting average:

ba <- sum(p_5[9:21])
ba
## [1] 0.93161

This batter is very impressive to have a 93% probability to exceed 0.400.

(c)

Given the distribution pX , what is his long-term batting average?

To get this we simply take the expected value of \(p_x\), since \(p_x\) gives the probability for each number of hits in a 4 at-bat sequence.

season_ba <- function(p){
  for(i in 1:length(p)){
    E = (i-1)*p[i]
  }
  return(E)
}
season_ba(p_x)
## [1] 0.4