Let x x1 x2 be independent identically distributed random

Question

Let X, X₁, X₂, ... be independent, identically distributed random variables, such that P(X = 0) = P(X = 1) = 1/2.

(a) Let N₁ be the number of 0's and 1's until the first appearance of the pattern 10. Find E N₁.

(b) Let N₂ be the number of 0's and 1's until the first appearance of the pattern 11. Find E N₂.

(c) Let N₃ be the number of 0's and 1's until the first appearance of the pattern 100. Find E N₃.

(d) Let N₄ be the number of 0's and 1's until the first appearance of the pattern 101. Find E N₄.

(e) Let N₅ be the number of 0's and 1's until the first appearance of the pattern 111. Find E N₅.

(f) Solve the same problem if X ∈ Be(p), for 0