Final answer:
The average time between two successive arrivals is 240 minutes. When the store first opens, it takes an infinite amount of time on average for three customers to arrive.
Step-by-step explanation:
a. To find the average time between two successive arrivals, we need to calculate the average of the time duration between each arrival. Since the deliveries are continuous and uniform, the time between two successive arrivals follows an exponential distribution.
We can use the exponential distribution formula: Average time between arrivals = 1 / Arrival rate. In this case, the arrival rate is the number of arrivals per hour. If we convert the delivery time from minutes to hours, we can say that the delivery time is (2 - 10) = 4 hours. So, the arrival rate is 1 / 4 = 0.25.
Hence, the average time between two successive arrivals is 1 / 0.25 = 4 hours, which is equivalent to 240 minutes.
b. To find the average time it takes for three customers to arrive when the store first opens, we need to consider the probability density function of the exponential distribution. The cumulative distribution function (CDF) of the exponential distribution can be used to find the probability of waiting less than or equal to a certain time.
The formula for the CDF is: CDF(t) = 1 - e^(-λt), where λ is the arrival rate and t is the waiting time. The average waiting time for three customers to arrive is the time at which the CDF reaches 3/3, or 1. By solving the equation 1 - e^(-λt) = 1, we find that e^(-λt) = 0, which implies that λt = ∞.
Therefore, the average time required for three customers to arrive when the store first opens is infinite.