Let $x = (x_0, \ldots, x_{n-1}) \in \{0,1\}^n$ and consider a noisy observation $$O(x) = \left(O_0(x), O_1(x), \ldots, O_{n-1}(x)\right)$$ of source $x$. a. If $0 \leq j \leq k \leq n-1$, show that $\mathbb{P}\left(N \geq j \text{ and } I_j = k\right) = p\binom{k}{j}p^j(1-p)^{k-j}$. b. Show that, for all $0 \leq j \leq n-1$, $$\mathbb{E}\left[O_j(x)\right] = p\sum_{k=j}^{n-1} x_k \binom{k}{j} p^j (1-p)^{k-j}.$$ c. Show that for all $\omega \in \mathbb{C}$, $$\mathbb{E}\left[\sum_{j=0}^{n-1} O_j(x) \omega^j\right] = p \sum_{k=0}^{n-1} x_k (p\omega + 1 - p)^k.$$
Let $x = (x_0, \ldots, x_{n-1}) \in \{0,1\}^n$ and consider a noisy observation
$$O(x) = \left(O_0(x), O_1(x), \ldots, O_{n-1}(x)\right)$$
of source $x$.
a. If $0 \leq j \leq k \leq n-1$, show that $\mathbb{P}\left(N \geq j \text{ and } I_j = k\right) = p\binom{k}{j}p^j(1-p)^{k-j}$.
b. Show that, for all $0 \leq j \leq n-1$,
$$\mathbb{E}\left[O_j(x)\right] = p\sum_{k=j}^{n-1} x_k \binom{k}{j} p^j (1-p)^{k-j}.$$
c. Show that for all $\omega \in \mathbb{C}$,
$$\mathbb{E}\left[\sum_{j=0}^{n-1} O_j(x) \omega^j\right] = p \sum_{k=0}^{n-1} x_k (p\omega + 1 - p)^k.$$