Question

Suppose it is known from large amounts of historical data that X, the number of cars that arrive at a specific intersection during a 20-second time period, is characterized by the following discrete probability function: f(x)=e−6 6x x! , for x =0 ,1,2,....
(a) Find the probability that in a specific 20-second time period, more than 8 cars arrive at the intersection. (b) Find the probability that only 2 cars arrive.

Answer 1

Let X be the number of cars that arrive at a specific intersection during a 20-second time period. Given the density function:

$$f(x)= \text{\textbraceleft}\begin{matrix} \dfrac{e^{-6}6^x}{x!}, for \ x=0, 1, 2, ... \\ 0, \ \ \ \ \ \ \ elsewhere \end{matrix}$$

a) We need to find the probability $P(X>8)$

$$P(X>8)=1-P(X \le8)=$$

$$1-{e^{-6}6^0 \over 0!}-{e^{-6}6^1 \over 1!}-{e^{-6}6^2 \over 2!}-{e^{-6}6^3 \over 3!}-{e^{-6}6^4 \over 4!}-{e^{-6}6^5 \over 5!}-{e^{-6}6^6 \over 6!}-{e^{-6}6^7 \over 7!}-{e^{-6}6^8 \over 8!}=$$

$$=1-341.8e^{-6} \approx 0.152766$$

b) We need to find the probability $P(X=2)$

$$P(X=2)={e^{-6}6^2 \over 2!}=18e^{-6}\approx0.044618$$