Four people working independently on an assembly line all perform the same task. The time (in minutes) to complete this task for person i (i = 1, 2, 3, 4) has a uniform distribution on the interval [0, i]. Suppose each person begins the task at the same time. (20 marks)
a. What is the probability that person 2 takes less than 90 seconds to complete the task?
b. What is the mean completion time for each person?
c. What is the probability that all four people complete the task in less than 30 seconds? (can use R to solve)
d. What is the probability that exactly one person completes the task in less than one minute? (can use R to solve)
To answer these questions, we'll need to calculate probabilities and means using the given information.
a. To calculate the probability that person 2 takes less than 90 seconds to complete the task, we need to determine the cumulative distribution function (CDF) for person 2's time.
The cumulative distribution function (CDF) is defined as the probability that a random variable takes a value less than or equal to a certain value. In this case, we need to find P(X ≤ 90) where X is the time taken by person 2.
Since the time taken by person 2 follows a uniform distribution on the interval [0, 2], the probability distribution function (PDF) is a horizontal line with height 1/2. The CDF is the integral of the PDF, which gives us the area under the curve up to a certain value.
By calculating the integral of the PDF from 0 to 90, we can find the probability that person 2 takes less than 90 seconds to complete the task.
b. To calculate the mean completion time for each person, we need to find the expected value or the average of their respective interval of time.
The mean of a uniform distribution on the interval [a, b] is given by (a + b) / 2. Since person i's time interval is [0, i], the mean completion time for person i would be (0 + i) / 2 = i / 2.
c. To calculate the probability that all four people complete the task in less than 30 seconds, we need to find the joint probability of each person completing the task within their respective time interval.
Since the completion times of each person are independent, we can multiply the individual probabilities together. For example, person 1 has a probability of 1 of completing the task within 0 seconds, person 2 within 2 seconds, person 3 within 3 seconds, and person 4 within 4 seconds. We can find the joint probability by multiplying these probabilities together.
d. To calculate the probability that exactly one person completes the task in less than one minute, we need to sum up the individual probabilities of each person completing the task within their respective time interval while the others take longer.
Let's separate this calculation into cases. We have four people, so there are four cases to consider: person 1, person 2, person 3, and person 4.
In each case, we need to calculate the probability of that person completing the task within their time interval (less than 1 minute) while the others take longer than 1 minute. We multiply this probability with the probabilities of the others taking longer (greater than 1 minute) and sum them up.
Now, armed with an understanding of these approaches, let's use R to find the exact solutions to the remaining questions.