Let $L(n)$ be a random walk defined as

$L(n)={\displaystyle \underset{i=1}{\overset{n}{}}}{Y}_i$

where

$Y_i=\{\begin{array}{cc}2,& {}_i=H\\ -3,& {}_i=T\end{array}$

and initial value $L(0)=0.$ For this walk answer the following:

$($a$)$ Assuming both temporal and spatial homogeneity, find the value of $$ where $=P({}_i=H)$ that makes this process a martingale.

$($b$)$ Express $=V_L^1(0,3)$

$($c$)$ Using the value for $$ that you found in part $($a$),$ determine $V().$