Question

The average number of claims per hour made to the State Insurance Company for
damages or losses incurred in moving is 3. What is the probability that in any given hour:
(i) Fewer than three claims will be made?
(ii) Exactly three claims will be made?
(iii) Three or more claims will be made?
(iv) More than three claims will be made?

Answer 1

This is the Poisson distribution with $\mu=3.$

(i) $P(X<3)=P(X=0)+P(X=1)+P(X=2)=e^{-\mu}(\frac{\mu^0}{0!}+\frac{\mu^1}{1!}+\frac{\mu^2}{2!})=0.4232.$

(ii) $P(X=3)=e^{-3}\frac{\mu^3}{3!}=0.2240.$

(iii) $P(X\ge3)=1-P(X<3)=1-0.4232=0.5768.$

(iv) $P(X>3)=1-P(X\le3)=1-P(X<3)-P(X=3)=1-0.4232-0.2240=$

$=0.3528.$