Question

The number of calls coming into an hotels reservation centre is poisson random variable with mean 3

i) Find the probability that no calls come in a given one minute period

ii)Assume that the number of calls arriving in two different minutes are independent. Find the probability that at least two calls will arrive in a given two minute period.

Answer 1

i) Let $X=$ the number of calls arriving  in a given one minute period: $X\sim Po(\lambda t)$

Given $\lambda=3, t=1$

$\lambda t=3(1)=3$

$$P(X=0)=\dfrac{e^{-3}\cdot3^0}{0!}=e^{-3}\approx0.049787$$

ii) Let $X=$ the number of calls arriving in a given two minute period: $X\sim Po(\lambda t)$

Given $\lambda=3, t=2$

$\lambda t=3(2)=6$

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

$$=1-\dfrac{e^{-6}\cdot6^0}{0!}-\dfrac{e^{-6}\cdot6^1}{1!}=1-7e^{-6}\approx0.982649$$