Final answer:
The mean waiting time for the arrival of the second customer is 1 minute and for the third customer, it's 1.5 minutes. The variance for these arrivals is 0.25 minutes² and 0.75 minutes² respectively. The probability density functions for these waiting times are exponential distributions with the corresponding means.
Step-by-step explanation:
Poisson Process and Exponential Distribution
Customers arriving at the checkout counter of a convenience store according to a Poisson process at a rate of two per minute can be analyzed using the properties of the exponential distribution. The mean or expected value (λ) for the time between arrivals is the reciprocal of the rate, which in this case is ½ minute (or 30 seconds) since customers arrive at a rate of two per minute.
Mean of Waiting Time: The mean waiting time for the arrival of the second customer, in this context, is the sum of the average waiting times for two customers which is 1 minute. For the third customer, it would be the sum for three customers, thus the mean waiting time would be 1.5 minutes (30 seconds × 3).
Variance of Waiting Time: The variance of the waiting time is equal to the square of the mean of the inter-arrival time. Thus, the variance for the arrival of both the second and third customers would be 0.25 minutes² and 0.75 minutes² respectively, as the inter-arrival times are independent and have the same exponential distribution.
Probability Density Function (PDF): The probability density function for the exponential distribution is given by f(t) = λ * e^{-λt}, where λ is the rate and t is the time. Here, the PDF for the waiting time until the arrival of the second and the third customer would be the convolution of two and three exponential distributions, which by a property of the Poisson process, would also be exponential distributions with means of 1 and 1.5 minutes, respectively.