FProblem: A student walks along a real line and tries to get to the origin. Each step he makes is random ; the larger the intended step, the greater the variance is of that step. When the student is at location $x$, the next step has a mean of $0$ and variance of $\alpha$. Let $X_n ={}$ position of the student after $n$ steps. Let $N\sim\mathrm{Poisson}(\lambda)$ Find (a) $E(X_N \mid X_0 = x_0)$ , (b) $\operatorname{Var}(X_N \mid X_0 = x_0)$ (Express $X_N$ as a sum of $N$ random variables)
Approach: I first expressed $X_N$ as sum of $N$ RVs : $X_N = X_1+ X_2 + \cdots+ X_N$. For (a) I am thinking $E(X_N \mid X_0 = x_0) = E(X_1+ X_2 + \cdots + X_N \mid X_0 = x_0) = n(0) = 0$ and don't know how to do (b). Am I on the right track?
$\newcommand{\var}{\operatorname{var}}\newcommand{\E}{\operatorname{E}}$I am left to surmise that the variance of every step is the same simply because of what hasn't been said, but explicitness about that point would normally be expected. Similarly the expected value of each step I would surmise is $0$. Finding $\var(X_1+\cdots+X_N)$ would be just a matter of adding the variances if $N$ were fixed, provided the steps are independent (which has also not been explicitly stated. \begin{align} \var(X_1+\cdots+X_N) & = \var(\E(X_1+\cdots+X_N\mid N)) + \E(\var(X_1+\cdots+X_N\mid N)) \\[8pt] & = \var(0) + \E( N\lambda) \\[8pt] & = 0 + \lambda^2 \E(N) = \lambda^2\cdot\lambda. \end{align}