Final answer:
The duration between visits to a website follows an exponential distribution with a mean of 12 visits per hour. Different probabilities related to the duration between visits can be calculated using the exponential and Poisson distribution formulas.
Step-by-step explanation:
The given scenario deals with the duration between visits to a website that follows an exponential distribution. In an exponential distribution, the time between events has a constant failure rate and is memoryless. Based on the information given, the correct answer would be:
a. Exponential distribution with mean 12
To find the probability that the duration between two successive visits is more than 10 minutes, we can use the exponential distribution formula: P(X > x) = e^(-λx), where λ is the rate parameter. In this case, λ = 1/12 visits per minute. Plugging in the values, we have P(X > 10) = e^(-1/12 * 10) = 0.4066.
b. To find the top 25 percent of durations between visits, we can use the exponential distribution formula: t = (-1/λ) * ln(1-p), where p is the desired percentile. In this case, p = 0.75, so t = (-1/(1/12)) * ln(1-0.75) = 1.7916.
c. To find the probability that the next visit will occur within the next five minutes, we can use the exponential distribution formula: P(X < x) = 1 - e^(-λx). Plugging in the values, we have P(X < 5) = 1 - e^(-1/12 * 5) = 0.3935.
d. To find the probability that fewer than seven visits occur within a one-hour period, we can use the Poisson distribution formula: P(X < x) = e^(-λ) * (λ^x / x!). In this case, λ = 12 visits per hour and x = 7. Plugging in the values, we have P(X < 7) = e^(-12) * (12^7 / 7!) = 0.0901.