Question

The number of claim per hour at the WBA Insurance Company has a Poisson distribution with mean 3 claim per hour. Find the probability that in any given hour there will be.

a)        exactly none claims  

b) exactly one claim                                                                                           

c)        exactly two claims                                                                                      

d)        three or more class                                                                                       

e)        exactly four claims in 2 hours

Answer 1

Let $X=$ the number of claims: $X\sim Po(\lambda t).$

Given $\lambda=3$

a)

$t=1$

$$P(X=0)=\dfrac{e^{-3(1)}\cdot(3(1))^0}{0!}=e^{-3}\approx0.049787$$

b)

$t=1$

$$P(X=1)=\dfrac{e^{-3(1)}\cdot(3(1))^1}{1!}=3e^{-3}\approx0.149361$$

c)

$t=1$

$$P(X=2)=\dfrac{e^{-3(1)}\cdot(3(1))^2}{2!}=4.5e^{-3}\approx0.224042$$

d)

$t=1$

$$P(X\geq3)=1-P(X=0)-P(X=1)$$

$$-P(X=2)=1-e^{-3}-3e^{-3}-4.5e^{-3}$$

$$=1-8.5e^{-3}\approx0.576810$$

e)

$t=2$

$$P(X=4)=\dfrac{e^{-3(2)}\cdot(3(2))^4}{4!}=54e^{-6}\approx0.133853$$