Question

Mrs Smith walks to work every day. If she takes the most direct route, her journey time is 10 minutes. However, this route follows a busy main road and, if it has been raining, she often gets splashed by traffic driving through puddles. Before she leaves for work therefore, she checks the weather: if the sky is no more than 40% covered by cloud, she takes the most direct route, but otherwise she takes a longer route where she is less likely to get splashed. This longer route takes her 15 minutes.

Suppose that the proportion of sky covered by cloud follows a beta distribution with pa- rameters α = 1/2, β = 1. What is Mrs Smith’s expected journey time? Compare this with her journey time if the cloud cover is equal to its expected value.

Answer 1