$6\mathrm{.}1.$ A fair die is thrown. Depending on the number of spots on the up face, the following random processes are generated. Sketch several examples of sample functions for each case. $(A$ is a constant.$)$

a$.x(t,)=\{\begin{array}{ccc}2A,& 1or2\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ 0,& 3or4\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ -2A,& 5or6\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ 3A,& 1\text{s}\text{p}\text{o}\text{t}& up\\ 2A,& 2\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ A,& 3\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ -A,& 4\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ -2A,& 5\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ -3A,& 6\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ 4A,& 1\text{s}\text{p}\text{o}\text{t}& up\\ 2A,& 2\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ At,& 3\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ -At,& 4\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ -2A,& 5\text{s}\text{p}\text{o}\text{t}\text{s}& up\\ -4A,& 6\text{s}\text{p}\text{o}\text{t}\text{s}& up\end{array}$

Section $6\mathrm{.}2$

$6\mathrm{.}2.$ Referring to Problem $6\mathrm{.}1,$ what are the following probabilities for each case?

a$.$