Final answer:
To solve the uniform distribution problem concerning train waiting times, we use the probability density function to calculate the probabilities for specific intervals and work with the cumulative distribution function for queries about percentages of commuters waiting longer than a certain time.
Step-by-step explanation:
The student's question involves a uniform distribution of waiting times for a train, which is a concept from probability and statistics within the field of mathematics. When working with uniform distributions, the probability density function (pdf) provides the likelihood of all outcomes within the range of possible values. In this case, the pdf is constant (f(x) = 1/40) for x between 3 and 43 minutes, which implies that the probability of waiting any specific length of time within that interval is the same.
To find the probability that a commuter waits between three and four minutes (part h), we calculate the area under the pdf in this interval. Since the distribution is uniform, this probability is equal to the length of the interval (4 minutes - 3 minutes = 1 minute) times the height of the pdf (1/40), which equals 1/40 or 0.025.
Meanwhile, part i requires expressing and solving a probability question, using the given knowledge that 60% of commuters wait more than a certain amount of time. To solve this, we would equate the cumulative distribution function (CDF) to 0.40 (since 1 - 0.60 = 0.40 for the lower 40% of the distribution) and solve for the time value.