Problem 31 - Calculus and its applications

Question

Duration of a phone call. A telephone company determines that the duration \(t,\) in minutes, of a phone call is an exponentially distributed random variable with a probability density function \(f(t)=2 e^{-2 t}, \quad 0 \leq t<\infty\) Find the probability that a phone call will last no more than 5 min.

Step-by-Step Solution

Answer

The probability is approximately 1, indicating it is almost certain that a call will last no more than 5 minutes.

Step 1: Understand the Probability Density Function

The given probability density function is exponential: \(f(t) = 2 e^{-2t}\) for \(0 \leq t < \infty\). This means the variable \(t\) follows an exponential distribution, which models the time until an event occurs, like the duration of phone calls in this case.

Step 2: Set up the Probability Expression

To find the probability that a phone call lasts no more than 5 minutes, we need to compute the probability \(P(t \leq 5)\). This is done by integrating the probability density function from \(0\) to \(5\).

Step 3: Integrate the Probability Density Function

Set up the integral: \( \int_{0}^{5} 2 e^{-2t} \, dt \). The integral will give us the cumulative probability of the event "call lasting no more than 5 minutes."

Step 4: Perform the Integration

Perform the integration: \[\int_{0}^{5} 2 e^{-2t} \, dt = \left[-e^{-2t}\right]_{0}^{5} \]Calculate this by first evaluating at the boundaries: \(-e^{-2(5)} - (-e^{-2(0)}) = -e^{-10} - (-1)\).

Step 5: Simplify the Expression

Simplify the result: \(1 - e^{-10}\). Since \(e^{-10}\) is a very small number close to 0, using a calculator may be necessary to find this value.

Step 6: Calculate the Final Probability

Calculate the numeric value of the expression: When computed, \(1 - e^{-10} \approx 1\). The probability that a phone call lasts no more than 5 minutes is almost certain, given that \(e^{-10}\) is nearly zero.

Key Concepts

Probability Density Function

In probability theory, the probability density function (PDF) is a key concept used to describe the distribution of continuous random variables. For a random variable to be exponentially distributed, like the duration of a phone call in this example, its PDF has a specific formula.
The exponential distribution is given by the function \( f(t) = 2 e^{-2t} \), for \( 0 \leq t < \infty \). This equation tells us that the most likely duration of a phone call decreases exponentially over time. The factor of 2 in the equation is known as the rate parameter, and it determines the rate at which the probability of longer durations decreases.

• The area under the PDF curve from \( t = 0 \) to \( t = \infty \) is 1, which satisfies the condition of being a true probability distribution.
• The PDF gives an idea of how the likelihood of various durations "density" peaks and tapers off as time increases.

Understanding the PDF helps us predict the behavior of the random variable and makes it easier to compute probabilities over certain intervals.

Integration

Integration is a fundamental concept when working with probability density functions. To find the probability that a random variable falls within a specific range, you need to integrate the PDF over that range. In our problem with the exponential distribution,
we compute the integral of the PDF from 0 to 5 minutes to find the probability of a call lasting no longer than 5 minutes.
The integration is set up as follows:\[ \int_{0}^{5} 2 e^{-2t} \, dt \]Performing this integral involves understanding the operand \( e^{-2t} \) and how it relates to the integration process.

• The integration utilizes the properties of exponential functions, which have a straightforward antiderivative.
• Upon integration, you evaluate the function at the boundary points to determine the cumulative probability.

Integration not only helps compute probabilities but also illustrates how cumulative probabilities are derived from the density function.

Cumulative Probability

Once integration is performed, the result provides the cumulative probability, which in this case represents the likelihood that the phone call ends in 5 minutes or less.
The cumulative probability is calculated as follows:
i. Compute the antiderivative during integration to get \[-e^{-2t}\],
ii. Evaluate it from \(0\) to \(5\), giving \(-e^{-10} + 1\).

• This leads to the final probability expression, \(1 - e^{-10}\), representing the cumulative probability.
• Given that \(e^{-10}\) is practically negligible, the probability that a phone call lasts no more than 5 minutes is almost 1, indicating a near-certain event.

Cumulative probabilities often offer more intuitive insights compared to the PDF alone, as they provide the total likelihood for a range of outcomes rather than just point-specific values.