The number of burglaries occurring on Burlington Street during a one-year period is Poisson distributed with mean 1.

Calculate the expected number of burglaries on Burlington Street in a one-year period, given that there are at least two burglaries.

• 0.63
• 2.39
• 2.54
• 3.00
• 3.78

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: B

Let X be the number of burglaries. Then,

[[math]] \begin{align*} \operatorname{E}(X | X \geq 2) = \frac{\sum_{x=2}^{\infty}xp(x)}{1-p(0)-p(1)} &= \frac{\sum_{x=0}^{\infty}xp(x)-p(0)-(1)p(1)}{1-p(0)-p(1)} \\ &= \frac{1-p(1)}{1-p(0)-p(1)} \\ &= \frac{1-e^{-1}}{1-e^{-1}-e^{-1}} = 2.39. \end{align*} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.