Suppose the number of cars passing through a toll booth in a 10 minute interval is a Poisson random variable. If the average number of cars is 23, give an expression for the probability that n cars pass through the booth. Is $p _ { n } = \frac { ( 23 ) ^ { n } } { n ! } e ^ { - 23 }$ correct?

Question

Suppose the number of cars passing through a toll booth in a 10 minute interval is a Poisson random variable. If the average number of cars is 23, give an expression for the probability that n cars pass through the booth. Is

p _ { n } = \frac { ( 23 ) ^ { n } } { n ! } e ^ { - 23 }

correct?

Quizplus · Accepted Answer

The given expression $p _ { n } = \frac { ( 23 ) ^ { n } } { n ! } e ^ { - 23 }$ correctly represents the probability of exactly $n$ cars passing through the toll booth in a 10-minute interval, according to the Poisson distribution formula $P(X = n) = \frac{\lambda^n e^{-\lambda}}{n!}$, where $\lambda = 23$ is the average rate (mean) of cars passing through per interval.

Suppose the Number of Cars Passing Through a Toll Booth pn=(23)nn!e−23p _ { n } = \frac { ( 23 ) ^ { n } } { n ! } e ^ { - 23 }pn​=n!(23)n​e−23

Suppose the Number of Cars Passing Through a Toll Booth $p _ { n } = \frac { ( 23 ) ^ { n } } { n ! } e ^ { - 23 }$