IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A random walk starts at 0 and each step is +1 with probability p = 0.6 and -1 with probability 0.4, independently. What is the expected position and the variance of the position after 50 steps?
The mean is 10 and the variance is 48. Each step has mean 0.2 and variance 1 minus 0.2 squared, which is 0.96. Because steps are independent, both quantities scale by 50, giving 10 and 48.
- AMean 10, variance 48Correct
- BMean 10, variance 50
- CMean 20, variance 48
- DMean 20, variance 50
- Mean 10, variance 24
Explanation
Each step has mean 0.6 - 0.4 = 0.2, so after 50 steps the mean is 10. Each step has E[Z^2] = 1, so the variance per step is 1 - 0.04 = 0.96. Over 50 steps the variance is 48. Using variance 50 ignores the squared mean.
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