John parks cars at a hotel. On the average, 6.0 cars will arrive in an hour. A driver's decision on whether to let John park the car does not depend upon any other person's decision. Define the random variable X to be the number of cars arriving in any hour period.
a. Wat is the appropriate probability distribution for X? Explain how X satisfies the properties of the distribution.
b. Compute the probability that exactly 5 cars will arrive in the next hour.]
c. Compute the probability that no more than 5 cars will arrive in the next hour.
a. The appropriate probability distribution for X is the Poisson distribution. The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, given the average rate of occurrence and the assumption of independence between events. In this case, X represents the number of cars arriving in an hour, and since the rate is given as an average of 6.0 cars per hour, we can use the Poisson distribution to model the probability of different numbers of cars arriving.
The properties of the Poisson distribution are:
1. The number of events occurring in disjoint intervals is independent of each other.
2. The average rate of occurrence is constant across the interval.
3. The probability of exactly k events occurring in the interval is given by the Poisson probability formula: P(X=k) = (e^(-λ) * λ^k) / k!, where λ is the average rate of occurrence and k is the number of events.
In this case, each car arriving is considered an event, and the number of cars arriving in an hour is the random variable X. The assumption that a driver's decision does not depend on others' decisions satisfies the independence property of the Poisson distribution.
b. The probability that exactly 5 cars will arrive in the next hour can be computed using the Poisson probability formula. We substitute λ=6.0 (average rate) and k=5 (number of events) into the formula, and calculate it as follows:
P(X=5) = (e^(-6.0) * 6.0^5) / 5!
c. To compute the probability that no more than 5 cars will arrive in the next hour, we can sum up the probabilities of X=0, X=1, X=2, X=3, X=4, and X=5 using the Poisson probability formula for each value of X, and add them together.