Problem 3 - Applied Calculus

Question

A person who travels regularly on the \(9: 00 \mathrm{am}\) bus from Oakland to San Francisco reports that the bus is almost always a few minutes late but rarely more than five minutes late. The bus is never more than two minutes early, although it is on very rare occasions a little early. (a) Sketch a density function, \(p(t)\), where \(t\) is the number of minutes that the bus is late. Shade the region under the graph between \(t=2\) minutes and \(t=4\) minutes. Explain what this region represents. (b) Now sketch the cumulative distribution function \(P(t)\). What measurement(s) on this graph correspond to the area shaded? What do the inflection point(s) on your graph of \(P\) correspond to on the graph of \(p ?\) Interpret the inflection points on the graph of \(P\) without referring to the graph of \(p\).

Step-by-Step Solution

Answer

The shaded area under \(p(t)\) from 2 to 4 minutes reflects the probability of these delay times. The CDF \(P(t)\)'s inflection indicates where the bus's lateness is most frequent.

Step 1: Understanding the Problem

Analyze the given problem to understand what is being asked. Part (a) of the problem requires a sketch of a probability density function (PDF), \(p(t)\), for the time \(t\) that the bus is late. We also need to indicate and interpret the area between \(t = 2\) and \(t = 4\). Part (b) asks for a sketch of the cumulative distribution function (CDF), \(P(t)\), and explore its relationship with the PDF, especially in terms of the shaded area and inflection points.

Step 2: Sketching the Density Function p(t)

Consider a plausible distribution based on the description. The bus being rarely early suggests a density starting near \(t = -2\) and mostly focused on being late. It peaks before\( t = 5\). A possible sketch is a right-skewed bell between \( -2\) to about \(5\), tapering after. The shaded area between \(t=2\) and \(t=4\) under the curve represents the probability that the bus is 2 to 4 minutes late.

Step 3: Interpreting the Shaded Area

The shaded region under \(p(t)\) between \(t = 2\) and \(t = 4\) represents \(P(4) - P(2)\), the probability of the bus being between 2 to 4 minutes late. It's an integral of \(p(t)\) between these two limits, indicating a frequency of delay within that time frame.

Step 4: Sketching the Cumulative Distribution Function P(t)

The CDF \(P(t)\) starts at 0 and increments positively, capturing the total probability from \(-2\) to \(t\). It increases more steeply between \(t=2\) and \(t=5\) where lateness is likeliest. The integration of these decisions vis-à-vis \( p(t) \) aligns with \( P(t) \)'s accelerated or decelerated ascent.

Step 5: Interpreting the Inflection Point of P(t)

In the CDF, an inflection point implies a change in the rate at which probability is accumulated. Where \(P(t)\) has an inflection point, \(p(t)\) is at a peak, reflecting the bus's most expected lateness periods. Here, it corresponds to \(t\), where \(p(t)\) shifts around the expected late time (common lateness peak).

Step 6: Final Interpretation and Summary

The density function captures a skewed distribution peaking near a common late period, and the shaded area indicates probabilities between 2 and 4 minutes, aligning with passenger insights. The cumulative plot reflects rising lateness accumulation probabilities proportionately, with critical understanding drawn from inflection overlaps.

Key Concepts

Cumulative Distribution Function

A Cumulative Distribution Function (CDF) provides a comprehensive way to understand probability distributions. It shows the probability that a random variable takes a value less than or equal to a specific number. In our context of the bus arrival time:

• The CDF, denoted as \(P(t)\), starts at zero, indicating zero probability at the minimum time, which in this case might be slightly early or exactly on time.
• It incrementally increases until it reaches 1, showing that the entire probability has been accounted for by the latest possible arrival time.
• The steepness of the CDF graph indicates how likely different times are. A steep slope between certain times, such as from 2 to 5 minutes late, suggests these are common outcomes.

Understanding the CDF helps in visualizing how probabilities are distributed over different times, with inflection points indicating the most probable outcomes within the distribution.

Probability Distribution

A Probability Distribution describes how the values of a random variable behave, often depicted through a sketch of a probability density function (PDF). This helps in understanding different outcomes and their probabilities.

• The PDF for our bus lateness starts slightly early, peaks near common late times, and tapers off towards rare late times.
• It is skewed towards lateness, illustrating the likelihood of lateness over being on time or early, echoing the everyday experience of travelers.
• The area under the PDF curve between specific times represents a cumulative probability, an essential concept for further analysis, such as in the CDF.

In this problem, focusing on the region between 2 and 4 minutes late under the PDF gives us the probability of arriving within that timeframe.

Sketching Graphs

Sketching graphs is a valuable skill for visualizing and understanding probability distributions. It allows for better interpretation of data and predictions about future outcomes. Here's how it applies to our exercise:

• For the PDF, sketching involves a start around a slightly early time and rising to a peak before tapering off. This visually shows the spread and skew of possible late times.
• In sketching the CDF, it begins at zero and climbs towards one, where the speed of its rise matches the commonness of different lateness times.
• Details such as shading the area under the PDF between 2 and 4 minutes help depict specific probabilities and further analyze the data collected.

Graphs ultimately act as visual tools to represent the probability metric that mathematical functions conceptualize, making them indispensable for intuitive understanding.