Question

In a town, 10 accident took place in 50 days assuming that the no. Of accidents follow Poisson distribution, find the probability that there will be 3 or more accidents in a day.

Answer 1

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

$$\lambda =\dfrac{\lambda t}{t} = \frac{10}{50} = 0.2$$

Distribution function:

$$P(X=k) = \frac{e^{-\lambda}\cdot\lambda^k}{k!} =\frac{ e^{-0.2}\cdot0.2^k}{k!}$$

Hence,

$$P(X \geq 3) = 1 - Pr(X < 3)$$

$$= 1 - P(X=0)-P(X=1)-P(X=2)$$

$$= 1 - e^{-0.2}(1 + \frac{0.2}{1} + \frac{0.2^2}{2}) = 0.00115$$

So, the probability is approximately 0.115%.