Question

The number of people arriving per hour at the emergency room (ER) of a local hospital seeking medical attention can be modeled by the Poisson distribution, with a mean of 14 people per hour. The inter-arrival time, XX, is defined as the time that passes between successive arrivals of patients seeking medical attention.

(a) How much time in minutes would you expect to pass between the arrival of successive patients seeking medical attention at this ER, in any given hour? Enter your answer to two decimals. If you need to use your answer in part (a) for the remaining parts of this question, use two decimals.

(b) A person has just arrived at the ER to receive medical attention. What is the probability that at most 7 minutes will pass until the next person seeking medical attention arrives? Use four decimals in your answer. If necessary, use your answer from a.

(c) It has been 4 minutes since the last person seeking medical attention arrived at the ER. What is the probability that at least 12 minutes (in total) will pass until the next medical-attention-seeking person passes through the ER doors? Use four decimals in your answer.

(d) 55% of the time, the inter-arrival time is at most how many minutes? Enter your answer to two-decimals.

Answer 1

The average time between successive arrivals can be found by converting the mean from people per hour to minutes per person. The probability that at most 7 minutes will pass until the next person arrives can be calculated using the exponential distribution formula. The probability that at least 12 minutes will pass until the next person arrives can be found by subtracting the probability of the opposite event from 1. The inter-arrival time that occurs at most 55% of the time can be determined by finding the corresponding quantile of the exponential distribution.

Step-by-step explanation:

(a) To find the average time between successive arrivals, we need to convert the mean from people per hour to minutes per person. There are 60 minutes in an hour, so we divide the mean by 60: 14/60 = 0.2333 minutes per person. Therefore, we would expect approximately 0.23 minutes to pass between the arrival of successive patients seeking medical attention at this ER in any given hour.

(b) To find the probability that at most 7 minutes will pass until the next person arrives, we can use the exponential distribution formula: P(X ≤ x) = 1 - e^(-λx), where λ is the rate parameter (in this case, the mean of 0.2333). Plugging in the values, we get P(X ≤ 7) = 1 - e^(-0.2333*7) ≈ 0.8293.

(c) To find the probability that at least 12 minutes will pass until the next person arrives, we can subtract the probability of the opposite event (that less than 12 minutes will pass) from 1. Using the same formula as in part (b), we get P(X > 12) = 1 - P(X ≤ 12) = 1 - (1 - e^(-0.2333*12)) ≈ 0.4720.

(d) To find the inter-arrival time that occurs at most 55% of the time, we need to find the corresponding quantile of the exponential distribution. In other words, we need to find the x-value such that P(X ≤ x) = 0.55. Using the exponential distribution formula and solving for x, we get x = -ln(1 - 0.55)/λ = -ln(0.45)/0.2333 ≈ 3.17 minutes.