A subway train on the Red Line arrives every eight minutes...

A subway train on the Red Line arrives every eight minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution. a. Define the random variable. $X=$ b. Graph the probability distribution. c. $f(x)=$ d. $\mu=$ e. $\sigma=$ f. Find the probability that the commuter waits less than one minute. g. Find the probability that the commuter waits between three and four minutes.

A subway train on the Red Line arrives every eight minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution.
a. Define the random variable. $X=$
b. Graph the probability distribution.
c. $f(x)=$
d. $\mu=$
e. $\sigma=$
f. Find the probability that the commuter waits less than one minute.
g. Find the probability that the commuter waits between three and four minutes.

Key Concepts

Standard Deviation of a Uniform Distribution

The standard deviation of a uniform distribution quantifies the amount of variation or dispersion from the mean. It is calculated using the formula (b - a)/?12 for a uniform distribution defined on the interval [a, b]. This measure helps in understanding how much the waiting times are spread out around the mean waiting time.

Mean of a Uniform Distribution

The mean (or expected value) of a uniform distribution is the midpoint of the interval, computed as (a + b)/2. It provides a measure of central tendency, indicating the average waiting time a commuter might expect when the arrival time is uniformly distributed across the interval.

Probability Calculations for Uniform Distribution

Probability calculations for a uniform distribution involve measuring the length of the subinterval of interest and dividing it by the total length of the interval. Because all outcomes are equally likely, the probability that a value falls within a specific interval is proportional to the size of that interval, reflecting the inherent uniformity of the distribution.

Uniform Distribution

The uniform distribution is a probability distribution where every outcome in a given interval has an equal probability of occurring. When modeling waiting times with a uniform distribution, it implies that the commuter is equally likely to arrive at any moment within the fixed time interval. It is defined by a lower bound and an upper bound, encapsulating the idea of complete randomness in the time interval.

Random Variable

A random variable is a variable whose value is subject to variations due to chance. It is defined within a probability framework and maps outcomes of a random process to numerical values. In the context of waiting times, the random variable represents the amount of time a commuter waits for the train, which can be measured continuously over an interval.

Probability Density Function

The probability density function (pdf) for a continuous random variable describes the relative likelihood for the variable to take on a given value. For a uniform distribution over an interval [a, b], the pdf is constant and equal to 1/(b - a) for all x in the interval. This constant value ensures that the total area under the curve over the interval equals one, which is a requirement for all probability distributions.

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