Let \(S = \{H, T\}^n\).
Let \(E_i = \{s \in S \) and either \(s_i = H\) or \( s_{i+1} = H\}\) for all \( 1 \leq i \leq n-1\).
Let \(R: S \to \mathbb{R}\) be the random variable defined by \[R = \sum_{i=1}^{n-1}\mathbf{1}_{E_i} \] be the sum of the indicator variables of \(E_i\).
(a) Briefly explain why \(\Pr [R = n-1] = F_{n+2}/2^n\). Use the fact that the size of \(C = \{s|s\in S \text{ for each }i, s_i = H \text{ or } s_{i+1}=H \}\) is \(F_{n+2}\).
If this looks familiar, it's because you've already done something similar in HW4
(b) Give with proof \(\mathbb{E} [R] \).
\(S=\{H, T\}^n\)
\(R = \sum_{i=1}^{n-1}\mathbf{1}_{E_i}\)
linearity of expectation
(b) Give with proof \(\mathbb{E} [R] \).
\(S=\{H, T\}^n\)
\(R = \sum_{i=1}^{n-1}\mathbf{1}_{E_i}\)
linearity of expectation
\(=\Pr [E_i]\)
the question remains to find \(\Pr[E_i]\)
(c) Obtain an upper bound on \(F_{n+2}\) using Markov's inequality.
\(S=\{H, T\}^n\)
(a) Give with proof the value of \(\text{Var}[R]\)
\(S=\{H, T\}^n\)