Please answer parts a) through d) and show work so I can learn. Thank you.

= 10.1. ]Let w1,w2, ... be considered as iid random variables on {H, T} such that P(wi P(wi = T) = 1/2. For each k e z>0 and for each w {H,T}N, define H) = 1, X (w): if wk = H 1-1, if wk = 1 { = Define a process (M1, M2, ...) by Mo = 0 and Mk D-X; for k > 0, and for each mez define Tm(w) to be the first positive integer l such that M (W) = m. (a) Show that for all n, k e z such that n > k, we have that Mn Mk has the same distribution as Mn-k. (b) Fix k e Z20. Define t to be the smallest value of n such that Mk+n Mk = 1. Show that has the same distribution as T1. (c) Show that for all n e z we have that M, and - M, have the same distribution. (d) Show that for all n e z we have that in and T-n have the same distribution. = 10.1. ]Let w1,w2, ... be considered as iid random variables on {H, T} such that P(wi P(wi = T) = 1/2. For each k e z>0 and for each w {H,T}N, define H) = 1, X (w): if wk = H 1-1, if wk = 1 { = Define a process (M1, M2, ...) by Mo = 0 and Mk D-X; for k > 0, and for each mez define Tm(w) to be the first positive integer l such that M (W) = m. (a) Show that for all n, k e z such that n > k, we have that Mn Mk has the same distribution as Mn-k. (b) Fix k e Z20. Define t to be the smallest value of n such that Mk+n Mk = 1. Show that has the same distribution as T1. (c) Show that for all n e z we have that M, and - M, have the same distribution. (d) Show that for all n e z we have that in and T-n have the same distribution