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Well, I don't see any way to get easy expressions. But for messy expressions, we have: $$E[Y] =\sum_{i=1}^n 2p_i(1-p_i) $$ Now suppose two strings differ by $y\geq 1$ bits. Enumerate these $\{1, \ld …

Let $\{X_i\}_{i=0}^{\infty}$ be an i.i.d. Bernoulli sequence with $P[X_i=1]=P[X_i=0]=1/2$. Consider: $$\{X_1, X_0, X_2, X_1, X_3, X_2, X_4, X_3, X_5, X_4, X_6, X_5, ...\}$$

The problem asked (without independence) can be solved. Fix nonnegative real numbers $a_1, …, a_n$. Define $M = \sum_{i=1}^n a_i$ and assume $M \geq 1$. Define $$\mathcal{X} = \{0, a_1\} \times \{0 …

It is interesting if you let the random index set depend on the realizations. For simplicity, restrict attention to random sequences $\{X_1, X_2, X_3, \ldots\}$ that converge to 0 in probability, but …

What about the standard Gram-Schmidt procedure? Given any collection of expectations $q_1, q_2, \ldots, q_n$, we can write: $$ X = q_1 + (q_2-q_1) + \cdots + (q_n-q_{n-1}) + (X-q_n) $$ Now define $ …

This answer is just spelling out what guest already said: If $p_k$ is iid over time $k \in \{0, 1, 2, \ldots\}$ then your system is equivalent to a discrete time homogeneous Markov chain with a fixed …

This looks like a "Wald equality" question. Define $Y=\sum_{i=1}^{T_n} X_i$. Then: \begin{align} 1 + x_{max} \geq Y =\sum_{i=1}^{\infty} X_i1\{T_n\geq i\}\\ \end{align} where $1\{T_n\geq i\}$ is a …

Let $\{X_1, X_2, \dotsc, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown param …