Problem 16 - Applied Calculus

Question

Which of the following functions makes the most sense as a model for the probability density representing the time (in minutes, starting from \(t=0\) ) that the next customer walks into a store? (a) \(p(t)=\left\\{\begin{array}{ll}\cos t & 0 \leq t \leq 2 \pi \\ e^{t-2 \pi} & t \geq 2 \pi\end{array}\right.\) (b) \(p(t)=3 e^{-3 t}\) for \(t \geq 0\) (c) \(p(t)=e^{-3 t}\) for \(t \geq 0\) (d) \(p(t)=1 / 4\) for \(0 \leq t \leq 4\)

Step-by-Step Solution

Answer

The most suitable function is (b) \(p(t)=3 e^{-3t}\).

Step 1: Understanding Probability Density Functions

A probability density function (PDF) must be non-negative and integrate to 1 over its domain. This is because probabilities range from 0 to 1, and the total probability of all outcomes in the sample space must be 1.

Step 2: Analyze Option (a)

The function \( p(t)=\left\{\begin{array}{ll}\cos t & 0 \leq t \leq 2 \pi \ e^{t-2 \pi} & t \geq 2 \pi\end{array}\right. \) cannot be a PDF. The cosine function \( \cos t \) is negative for some intervals within \( 0 \leq t \leq 2\pi \), and the piece for \( t \geq 2\pi \) does not contribute to the total probability integrating to 1. Hence, this option is not valid.

Step 3: Analyze Option (b)

The function \( p(t)=3 e^{-3 t} \) for \( t \geq 0 \) is exponential and integrates from 0 to infinity, potentially making it a valid PDF for non-negative \( t \). Integrating \( 3 e^{-3t} \) over its domain should yield 1. For confirmation:\[\int_{0}^{\infty} 3 e^{-3t}\, dt = \left[-e^{-3t}\right]_0^\infty = 1\]Thus, this function is a valid PDF.

Step 4: Analyze Option (c)

The function \( p(t)=e^{-3 t} \) for \( t \geq 0 \) is also an exponential function. To verify if it is a PDF, integrate over its domain:\[\int_{0}^{\infty} e^{-3t}\, dt = \left[-\frac{1}{3}e^{-3t}\right]_0^\infty = \frac{1}{3}\]This integration does not equal 1, so this is not a valid PDF.

Step 5: Analyze Option (d)

The function \( p(t)=1 / 4 \) for \( 0 \leq t \leq 4 \) is constant over a finite interval and should be evaluated as follows:\[\int_{0}^{4} \frac{1}{4}\, dt = \left[\frac{1}{4}t\right]_0^4 = 1\]This integrates to 1 over its domain, making it a valid PDF.

Step 6: Comparing the Valid Options

Both Option (b) with \( p(t)=3 e^{-3 t} \) and Option (d) with \( p(t)=1/4 \) are valid PDFs. Generally, exponential distributions are more suitable for modeling time between events, such as the time until the next customer enters a store. Therefore, Option (b) is the most sensible model.

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution that often models time until an event occurs. It is particularly useful for representing the time between independent events that happen at a constant average rate. For example, it can predict how long you'll have to wait for the next bus or the next customer to enter a store.

Key characteristics of an exponential distribution include:

• The distribution only takes on non-negative values. This is because time cannot be negative.
• The parameter \( \lambda \), which is the rate at which events occur. Higher values of \( \lambda \) indicate faster occurrences of events.
• The memoryless property, meaning the probability of an event occurring in the future is independent of any past events.

A fundamental property of the exponential distribution is its probability density function (PDF), represented as \( p(t) = \lambda e^{-\lambda t} \) for \( t \geq 0 \). This PDF signifies the likelihood of events happening at time \( t \). An important aspect here is ensuring that the PDF integrates to one, confirming it is a valid distribution model.

Integration

Integration is a core mathematical concept used to determine areas under curves, among other applications. In probability, integrating a probability density function over its domain verifies its validity by ensuring the total probability equals one. This concept ensures that the probabilities assigned to all possible outcomes in the sample space are accounted for.

In the context of probability density functions, potentially useful operations include:

• Definite integration, which calculates the total probability over a specific interval. For valid PDFs on infinite domains, such as with exponential distributions, we expect integration to naturally sum to one.
• Using substitution or tables to integrate complex functions seamlessly.

For example, for the exponential distribution option in the exercise, the integration \( \int_{0}^{\infty} 3e^{-3t}\, dt \) yields \( 1 \), confirming that \( p(t) = 3e^{-3t} \) is indeed a valid PDF. Understanding and mastering integration is essential when working with probability models as it helps verify and explain probability distribution behavior.

Probability Models

Probability models are mathematical representations that describe random phenomena. They are essential in predicting and understanding the behavior of complex systems based on probability theory. Probability models help make informed decisions in uncertain situations by estimating the likelihoods of various outcomes.

Some features of probability models include:

• Empirically derived models that rely on actual data and observed frequencies to form probabilities.
• Synthetically derived models, which use theoretical principles and assumptions for predictions.
• Often include both discrete and continuous types. Discrete probability models deal with countable outcomes, whereas continuous models deal with outcomes that can take any value in a range.

In the exercise context, different functions were considered to model the probability density of customer arrivals. The exponential distribution was determined to be the most appropriate model due to its effectiveness in representing time until events. By understanding and applying probability models, we can make meaningful conclusions about real-world situations and improve decision-making processes.