Question

Let metro trains on a certain line run every 10 minutes between midnight and 6 am. Let X denote the waiting time (in minutes) for the next train, under the assumption that a person arrives randomly during this time. X follows a uniform distribution between 0 and 10 minutes.

Answer 1

The student is asking about the waiting time for a metro train, which is uniformly distributed between 0 and 10 minutes, denoted as X~U(0, 10). The probability density function for this uniform distribution is f(x) = 1/10 for 0 ≤ x ≤ 10.

Step-by-step explanation:

The student's question pertains to the waiting time for a metro train, which is modeled by a uniform distribution. Specifically, we want to define a random variable X that represents the waiting time for the next train when trains run every 10 minutes. Since the trains run at regular intervals, and a person arrives randomly during this interval, the time X that a person must wait is indeed uniformly distributed between 0 and 10 minutes. This can be expressed as X~U(0, 10), where U stands for the uniform distribution.

The probability density function for a uniform distribution in the interval [a,b] is f(x) = 1/(b-a) for a ≤ x ≤ b. In the case of the metro train example, this would be f(x) = 1/10 for 0 ≤ x ≤ 10. To find specific probabilities, such as the probability of waiting less than a certain amount of minutes, you would integrate the density function over the desired interval.

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