[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

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

[[math]] p_X = \pmatrix{ 0 & 1 & 2 & 3 & 4 \cr .4 & .2 & .2 & .1 & .1\cr}. [[/math]]

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

• Let [math]X[/math] denote the number of hits that he gets in a series. Using the program NFoldConvolution, find the distribution of [math]X[/math] for each of the possible series lengths: four-game, five-game, six-game, seven-game.
• 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.)
• Given the distribution [math]p_X[/math], what is his long-term batting average?