Question

In a town 10 accidents took place in a period of 50 days. Assume that the number of accidents per day follow Poisson distribution. Find the probability that there will be three or more accidents per day.

Answer 1

For random variable from Poisson distribution with parameter $\lambda$ :

$\lambda = EX = \frac{10}{50} = 0.2$ , where $EX$ is sample mean

Distribution function:

$Pr(X=k) = e^{-\lambda}\frac{\lambda^k}{k!} = e^{-0.2}\frac{0.2^k}{k!}$

Hence,

$Pr(X \geq 3) = 1 - Pr(X < 3) = 1 - \sum_{k=0}^{2}Pr(X=k) = 1 - e^{-0.2}(1 + \frac{0.2}{1} + \frac{0.2^2}{2}) = 0.0011$

So, the probability is approximately 0.11%