Question

(b) Let $(X_n)_{n\ge 0}$ be a simple symmetric random walk on the integers starting at $x \in \mathbb{Z}$, that is, $X_n = \begin{cases} x & \text{if } n = 0\\ x + \sum_{i=1}^n Y_i & \text{if } n \ge 1 \end{cases}$, where $(Y_n)_{n\ge 1}$ is a sequence of IID random variables with $P(Y_n = 1) = P(Y_n = -1) = \frac{1}{2}$. Let $T = \min\{n \ge 0 : X_n = 0\}$ be the time that the walk first hits 0. (i) Let $n$ be a positive integer. For $0 < x < n$, calculate the probability that the walk hits 0 before it hits $n$. (ii) Let $x = 1$ and let $A$ be the event that the walk hits 0 before it hits 3. Find $P(X_1 = 0|A)$. Hence find $E(T|A)$. (iii) Let $x = 1$ and let $B$ be the event that the walk hits 0 before it hits 4. Find $E(T|B)$.

          (b) Let $(X_n)_{n\ge 0}$ be a simple symmetric random walk on the integers starting at $x \in \mathbb{Z}$, that is,
$X_n = \begin{cases} x & \text{if } n = 0\\ x + \sum_{i=1}^n Y_i & \text{if } n \ge 1 \end{cases}$,
where $(Y_n)_{n\ge 1}$ is a sequence of IID random variables with $P(Y_n = 1) = P(Y_n = -1) = \frac{1}{2}$.
Let $T = \min\{n \ge 0 : X_n = 0\}$ be the time that the walk first hits 0.
(i) Let $n$ be a positive integer. For $0 < x < n$, calculate the probability that
the walk hits 0 before it hits $n$.
(ii) Let $x = 1$ and let $A$ be the event that the walk hits 0 before it hits 3. Find
$P(X_1 = 0|A)$. Hence find $E(T|A)$.
(iii) Let $x = 1$ and let $B$ be the event that the walk hits 0 before it hits 4. Find
$E(T|B)$.

$(b) Let (Xn)n≥ 0 be a simple symmetric random walk on the integers starting at x ∈ℤ, that is, Xn = x if n = 0 x + ∑i=1^n Yi if n ≥ 1, where (Yn)n≥ 1 is a sequence of IID random variables with P(Yn = 1) = P(Yn = -1) = (1)/(2). Let T = min{n ≥ 0 : Xn = 0} be the time that the walk first hits 0. (i) Let n be a positive integer. For 0 < x < n, calculate the probability that the walk hits 0 before it hits n. (ii) Let x = 1 and let A be the event that the walk hits 0 before it hits 3. Find P(X1 = 0|A). Hence find E(T|A). (iii) Let x = 1 and let B be the event that the walk hits 0 before it hits 4. Find E(T|B).$

Added by Ana S.

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