Answer:
[tex] P(X=3) = 2.5^3 \frac{e^{-2.5}}{3!}=0.214[/tex]
Step-by-step explanation:
Definitions and concepts
The Poisson process is useful when we want to analyze the probability of ocurrence of an event in a time specified. The probability distribution for a random variable X following the Poisson distribution is given by:
[tex]P(X=x) =\lambda^x \frac{e^{-\lambda}}{x!}[/tex]
And the parameter [tex]\lambda[/tex] represent the average ocurrence rate per unit of time
For this case we know that [tex] \lambda =0.5 cars/min[/tex]
And we want the probability that during the next five minutes, three cars will arrive, so then our rate becomes:
[tex] \lambda = 0.5 \frac{cars}{min}* 5 min = 2.5 cars[/tex]
And we want this probability:
[tex] P(X=3)[/tex]
And if we use the pmf we got:
[tex] P(X=3) = 2.5^3 \frac{e^{-2.5}}{3!}=0.214[/tex]
