Can someone help me with how to solve this, please?
A bus comes by every 10 minutes. The times from when a person arives at the busstop until the bus arrives follows a Uniform distribution from 0 to 10 minutes. A person arrives at the bus stop at a randomly selected time. Round to 4 decimal places where possible.
The mean of this distribution is
.
The standard deviation is
.
The probability that the person will wait more than 6 minutes is
.
Suppose that the person has already been waiting for 2.4 minutes. Find the probability that the person's total waiting time will be between 4.7 and 6.8 minutes.
15% of all customers wait at least how long for the train?
minutes
To solve these questions, we need to use the properties of a Uniform distribution and some basic probability concepts. Let's go step by step.
1. The mean of a Uniform distribution is simply the average of the minimum and maximum values. In this case, the minimum value is 0 minutes and the maximum value is 10 minutes. So, the mean is (0 + 10) / 2 = 5 minutes.
2. The standard deviation of a Uniform distribution can be calculated using the formula: (√(b^2 - a^2)) / √12, where 'a' is the minimum value and 'b' is the maximum value. In this case, a = 0 and b = 10. So, the standard deviation is (√(10^2 - 0^2)) / √12 = 10 / √12 = 2.8868 (rounded to 4 decimal places).
3. To find the probability that the person will wait more than 6 minutes, we need to calculate the area under the probability density function (PDF) curve for the interval (6, 10]. Since the distribution is Uniform, the PDF is simply a rectangle with base 10 and height 1/10 (because the total area under the curve is 1). So, the probability is the area of the rectangle from 6 to 10, which is (10 - 6) * (1/10) = 4/10 = 0.4.
4. Suppose the person has already been waiting for 2.4 minutes. We want to find the probability that the total waiting time will be between 4.7 and 6.8 minutes. We can subtract the probability of waiting less than 4.7 minutes from the probability of waiting less than 6.8 minutes to get the desired probability. To calculate the probability of waiting less than a certain time, we need to find the area under the PDF curve up to that time. So, the probability is (6.8 - 0) * (1/10) - (4.7 - 0) * (1/10) = 0.68 - 0.47 = 0.21.
5. To determine the percentage of customers who wait at least a certain time, we need to find the area under the PDF curve from that time to the maximum time (10 minutes in this case). So, we want to find the area from 'x' to 10 and calculate it as (10 - x) * (1/10) = (10 - x) / 10. We are given that this probability is 15%, so we set up the equation (10 - x) / 10 = 0.15 and solve for 'x'. Cross-multiplying, we get 10 - x = 0.15 * 10, which simplifies to x = 10 - 1.5 = 8.5 minutes.
So, 15% of all customers wait at least 8.5 minutes for the bus.
Remember to round your answers to 4 decimal places where possible.