Final answer:
To calculate the probabilities for the given exponential probability distribution with a mean of 8 minutes per customer, we use the formulas and techniques for the exponential distribution. The probabilities for different ranges of time between events are calculated using the cumulative distribution function.
Step-by-step explanation:
The exponential probability distribution is a continuous probability distribution that models the time between events that occur at a constant rate. In this case, the mean of the distribution is 8 minutes per customer. To calculate the probabilities:
a) P(x > 13): This is the probability that the time between events is greater than 13 minutes. Since the exponential distribution is memoryless, this is equal to 1 - P(x <= 13) = 1 - e^(-8*13) ≈ 0.0337.
b) P(x > 3): This is the probability that the time between events is greater than 3 minutes. Using the same formula as before, we get 1 - e^(-8*3) ≈ 0.982.
c) P(8 <= x <= 19): This is the probability that the time between events is between 8 and 19 minutes. Using the cumulative distribution function, we have P(x <= 8) = 1 - e^(-8*8) ≈ 0.632 and P(x <= 19) = 1 - e^(-8*19) ≈ 0.877. Therefore, P(8 <= x <= 19) = P(x <= 19) - P(x <= 8) ≈ 0.877 - 0.632 ≈ 0.245.
d) P(1 <= x <= 6): This is the probability that the time between events is between 1 and 6 minutes. Using the same approach as before, P(x <= 1) = 1 - e^(-8*1) ≈ 0.993 and P(x <= 6) = 1 - e^(-8*6) ≈ 0.999. Therefore, P(1 <= x <= 6) = P(x <= 6) - P(x <= 1) ≈ 0.999 - 0.993 ≈ 0.006.