Let p=1/2 and let B$_n$ = 1 if {w$_n$=H} and B$_n$ = -1 if {w$_n$=T} where w$_n$ represents the outcome of a coin flip at time n.
Given that A$_n$ = B$_n$+B$_{n-1}$+B$_{n-2}$ , calculate E$_3$[A$_4^2$](T,H,H)
My attempt:
A$_4$ = B$_4$+B$_{3}$+B$_{2}$ = B$_4$ + 1 + 1 = B$_4$ + 2
E$_3$[A$_4^2$](T,H,H) = E$_3$[(B$_4$+2)$^2$] =
E$_3$[B$_4^2$+4B$_4$+4]=E$_3$[B$_4^2$]+4E$_3$[B$_4$]+4=
E$_3$[B$_4$]E$_3$[B$_4$] + 4E$_3$[B$_4$] + 4 =
[1(1/2)+(-1)(1/2)][1(1/2)+(-1)(1/2)]+[1(1/2)+(-1)(1/2)]+4 = 0+0+4 = 4
I don't think I can split E$_3$[B$_4^2$] because the expectation of the products of two independent variables is the product of the expectations, but B$_4^2$ is dependent on itself I believe? Any suggestions?