Question

The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution
with a mean of 5. Interest centers around the time that
elapses before 10 automobiles appear at the intersection.
(a) What is the probability that more than 10 automobiles appear at the intersection during any given
minute of time?
(b) What is the probability that more than 2 minutes
elapse before 10 cars arrive?

Answer 1

Answer:

a) 0.0137

b) 0.4579

Step-by-step explanation:

Let "X" represents the number of automobiles that arrive at a certain intersection per minute

a) $P(X>10)$ $= 1- P(X\leq 10)$

$1-\frac{e^a* a^X}{a!}$

$= 1- [P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)]\\= 1-0.0067-0.0377-0.0842-0.1404-0.1755-0.1755-0.1462-0.1044-0.0653-0.0363-0.0181\\= 1-0.9863\\= 0.0137$

b) $P(X>2)$

$P(X>2) = 1-P(X\leq 2)$$= 1-P(Y\leq 2)\\= 1-\int\limits^2_0 {\frac{x^{\alpha-1} * e^\frac{-x}{\beta }  }{\beta^\alpha * 9! } } \, dx \\= 1-\int\limits^{10}_0 {\frac{y^9* e^{-y}}{9!} } \, dx \\= 1-0.5421\\= 0.4579$