Answer to Question #257559 in Statistics and Probability for pierre

Question

The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean λ=7.

​(a) Compute the probability that more than 9 customers will arrive in a 2​-hour period.

​(b) What is the mean number of arrivals during a 2​-hour ​period?

Answer 1

Let $X=$ the number of customers arriving per period $t$ : $X\sim Po(\lambda t).$

(a) Given $\lambda=7, t=2\ h$

$\lambda t=7(2)=14$

$$P(X>9)=1-P(X\leq9)$$

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

$$-P(X=3)-P(X=4)-P(X=5)$$

$$-P(X=6)-P(X=7)-P(X=8)$$

$$-P(X=9)=1-\dfrac{e^{-14}\cdot14^0}{0!}-\dfrac{e^{-14}\cdot14^1}{1!}$$

$$-\dfrac{e^{-14}\cdot14^2}{2!}-\dfrac{e^{-14}\cdot14^3}{3!}-\dfrac{e^{-14}\cdot14^4}{4!}$$

$$-\dfrac{e^{-14}\cdot14^5}{5!}-\dfrac{e^{-14}\cdot14^6}{6!}-\dfrac{e^{-14}\cdot14^7}{7!}$$

$$-\dfrac{e^{-14}\cdot14^8}{8!}-\dfrac{e^{-14}\cdot14^9}{9!}=0.89060063$$

(b)

$$mean=\lambda t=7(2)=14$$

The mean number of arrivals during a 2​-hour ​period is $14.$