Problem 199 - Applied Statistics and Probability for Engineers

Question

When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each caller orders one ticket. What is the probability that the bus is filled in less than three hours from the time of the fare reduction?

Step-by-Step Solution

Answer

The probability that the bus is filled in less than three hours is approximately 0.747.

Step 1: Understand the Problem

We need to find the probability that 4 passengers (due to 4 available seats) purchase tickets in less than 3 hours after fare reduction. The time between ticket requests is exponentially distributed with a mean of 30 minutes.

Step 2: Identify the Distribution and Parameters

The exponential distribution is defined by its rate parameter (\(\lambda\)) which is the reciprocal of the mean. Here, the mean is 30 minutes, so \(\lambda = \frac{1}{30} \text{ tickets/minute}\).

Step 3: Setup the Time Condition

We are interested in the time for 4 requests, which follows a gamma distribution because the sum of exponential random variables with the same rate parameter follows a gamma distribution. However, since it is the sum of 4 identical exponential distributions, it is equivalent to an Erlang distribution, which is a special case of the gamma distribution.

Step 4: Calculate the Required Probability

The sum of 4 exponential random variables (each with mean 30 mins) has the parameter \(\lambda=\frac{1}{30}\). We seek the cumulative probability that this sum (or equivalently, the time taken to fill the bus) is less than 180 minutes (3 hours). This is calculated by finding \(P(T < 180)\), where \(T\) is a gamma(4,30) distributed random variable:\[ P(T<180) = 1 - \sum_{k=0}^{3} \frac{\left(\lambda t\right)^k e^{-\lambda t}}{k!} \]With \(\lambda = \frac{1}{30}\) and \(t = 180\), we compute this using mathematical tools or tables.

Step 5: Simplified Computation with Erlang (Gamma) Distribution

With a mean arrival time of 30 mins and 4 passengers, the sum is equivalent to Erlang(4, \(\frac{1}{30}\)). For practical computation, use the Erlang CDF from statistical software or tables for \(k=4, \lambda= \frac{1}{30}, t=180\) to find \(P(T < 180)\).

Key Concepts

Exponential Distribution

The exponential distribution is a fundamental probability distribution in statistics, widely used to model the time between events in a Poisson process. It's characterized by a constant average rate at which events occur. In simpler terms, if you are waiting for an event that's expected to happen on average every 30 minutes, an exponential distribution can describe whether it will happen sooner or later than expected.

Key features of the exponential distribution include:

• The rate parameter, denoted by \(\lambda\), which is the reciprocal of the mean. For instance, if the mean time between events is 30 minutes, as in the exercise, then \(\lambda = \frac{1}{30}\).
• The distribution is memoryless, which means the probability of an event occurring in the next period of time is the same, regardless of how much time has already elapsed.
• It's unimodal, having its peak at zero, showing the most likely time for the next event to happen immediately after the last.

Understanding the exponential distribution is essential in calculating probabilities involving time intervals, such as the likelihood of ticket sales in this scenario.

Erlang Distribution

The Erlang distribution is closely related to the exponential distribution but involves a series of events rather than just one. It is specifically useful when dealing with multiple, sequential occurrences of events. Imagine you are interested in the total time taken for a series of four ticket calls. You could use the Erlang distribution to model this when each call is exponentially distributed as in our bus scenario.

Characteristics of the Erlang distribution include:

• It is a special case of the gamma distribution where the shape parameter \(k\) (the number of events) is an integer.
• The rate parameter \(\lambda\) remains the same across the events, as each event is exponentially distributed.
• The Erlang distribution helps in calculating the total time required for all events to occur. For our bus problem, we consider 4 passengers getting tickets, modeled as an Erlang (4, \(\frac{1}{30}\)).

The Erlang distribution, therefore, becomes integral when predicting the probability of filling the bus within a fixed time frame.

Gamma Distribution

The gamma distribution generalizes the exponential and Erlang distributions. It's versatile, helping model scenarios where waiting times or life spans need to be considered. In cases where events need not be simply sequential, such as finding the probability for aggregated time for multiple tickets, this distribution provides the solution.

The gamma distribution is characterized by:

• A shape parameter \(k\), which need not be an integer contrary to the Erlang distribution.
• A scale parameter \(\theta = \frac{1}{\lambda}\), a reflection of how widely values are spread around the mean.
• It's crucial in calculating confidence intervals for variance and in Bayesian statistics.

By considering the bus ticket example in the context of gamma distribution, the problem involves finding the probability of an aggregated event time without simplifying assumptions needed for Erlang.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) is a key concept that allows us to calculate the probability that a random variable takes on a value less than or equal to a particular number. For our problem involving exponential, Erlang, and gamma distributions, the CDF becomes significant.

Some vital aspects of the CDF are:

• It starts at 0 and approaches 1 as the variable increases, showing accumulated probability.
• In many cases, it must be calculated through mathematical tools as it involves integration of the probability density function (PDF).
• For practical computations with the Erlang distribution, like in our problem, statistical software is used to estimate CDF values to find the probability within desired limits.

Utilizing CDF allows us to easily compute the probability that a given number of ticket purchases occurs in a set timeframe, such as under three hours, making it essential in our bus example.