Question

A short term insurance company receives three motor vehicle claims, on average, per day. Assume that the daily claims follow a Poisson process.

Required:

a) What is the probability that at most two motor vehicle claims are received in any given day?

b) What is the probability that more than two motor vehicle claims are received in any given period

of two days?

Answer 1

We have a Poisson distribution,

$λ=3;\\ P_t(X=k)=\cfrac{(\lambda t)^k\cdot e^{-\lambda t}}{k!}=\cfrac{(3t)^k\cdot e^{-3t}}{k!}.\\ \text{a) } P_1(X\le2)=\\ =P_1(X=0)+P_1(X=1)+P_1(X=2)=\\ =\cfrac{(3\cdot1)^0\cdot e^{-3\cdot1}}{0!}+\cfrac{(3\cdot1)^1\cdot e^{-3\cdot1}}{1!}+\\ +\cfrac{(3\cdot1)^2\cdot e^{-3\cdot1}}{2!}=\\ =0.0498+0.1494+0.2240=0.4232.$

$\text{b) } P_2(X>2)=1-P_2(X\le2)=\\ =1-(P_2(X=0)+P_2(X=1)+P_2(X=2)) ;\\ P_2(X=0)=\cfrac{(3\cdot2)^0\cdot e^{-3\cdot2}}{0!}=0.0025;\\ P_2(X=1)=\cfrac{(3\cdot2)^1\cdot e^{-3\cdot2}}{1!}=0.0149;\\ P_2(X=2)=\cfrac{(3\cdot2)^2\cdot e^{-3\cdot2}}{2!}=0.0446;\\ P_2(X>2)=1-(0.0025+0.0149+0.0446)=0.9380.$