Final answer:
a. The average time between two successive calls is 15 seconds. b. The probability that the next call occurs in less than 10 seconds after a call is received is 0.4866. c. The probability that exactly five calls occur within a minute is 0.1563.
Step-by-step explanation:
a. To find the average time between two successive calls, we need to determine the reciprocal of the call rate, which is the mean of the exponential distribution. In this case, the call rate is 4 calls per minute, so the average time between calls is 1/4 minutes or 15 seconds.
b. The probability that the next call occurs in less than 10 seconds after a call is received can be found by calculating the cumulative distribution function (CDF) of the exponential distribution. Using the exponential CDF formula, we can find the probability as P(X<10) = 1 - e^(-λt), where λ is the rate and t is the time. Substituting the values, we get P(X<10) = 1 - e^(-4*(10/60)) = 1 - e^(-2/3) = 0.4866.
c. The probability that exactly five calls occur within a minute can be calculated using the Poisson distribution formula. The Poisson probability mass function (PMF) is P(X=k) = (e^(-λ) * λ^k) / k!, where λ is the mean and k is the number of occurrences. In this case, λ=4 and k=5, so substituting the values, we get P(X=5) = (e^(-4) * 4^5) / 5! = 0.1563.
d. The probability that fewer than five calls occur within a minute can be calculated by summing the probabilities of 0, 1, 2, 3, and 4 calls using the Poisson distribution formula. P(X<5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4). Calculate each probability using the formula P(X=k) = (e^(-λ) * λ^k) / k! and sum them up.
e. The probability that more than 40 calls occur in an eight-minute period can be calculated using the Poisson distribution. However, since the number of calls per minute has a Poisson distribution, we need to convert the time period from eight minutes to minutes per minute. Multiply 40 by eight to get 320 calls and then use the Poisson distribution formula to calculate the probability of more than 320 calls within eight minutes.