IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes
A simple random walk starts at 0. At each step it moves +1 with probability p and -1 with probability q = 1 - p, with steps independent. Which statement about E[X_n] and Var[X_n] after n steps is correct?
Each step has mean p minus q and variance 4pq, because the step takes values +1 and -1, which are two units apart. Over n independent steps the mean is n(p - q) and the variance is 4npq.
- AE[X_n] = n(p - q) and Var[X_n] = 4npqCorrect
- BE[X_n] = n(p - q) and Var[X_n] = npq
- CE[X_n] = np and Var[X_n] = npq
- DE[X_n] = n(p + q) and Var[X_n] = 4npq
- E[X_n] = 0 and Var[X_n] = n
Explanation
Each step Y has mean p - q and E[Y^2] = 1, so Var[Y] = 1 - (p-q)^2 = (p+q)^2 - (p-q)^2 = 4pq. Over n independent steps the mean is n(p-q) and the variance is 4npq. Using npq forgets that the step size spans 2 units (from -1 to +1).
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