Problem 11 - Applied Calculus

Question

A person who travels regularly on the 9: 00 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

Shade between 2 and 4 minutes on \( p(t) \); it represents that time's probability. In \( P(t) \), the change from \( P(2) \) to \( P(4) \) is this area. Inflection points show change rates in probability density.

Step 1: Understanding a Density Function

A density function, denoted as \( p(t) \), represents how frequency is distributed over a range of outcomes (in this case, the lateness of the bus). The area under the curve of this function between two points indicates the probability that a value falls between these two points. Let's assume \( t=0 \) corresponds to the bus being exactly on time. The function should be skewed towards positive \( t \) since the bus is mostly a few minutes late but not more than 5 minutes.

Step 2: Sketching the Density Function

To sketch \( p(t) \), draw a horizontal axis representing time \( t \) (in minutes) and a vertical axis for the probability density. From \( t = -2 \) to \( t = 0 \) (early range), the function should have low values since it's rare for the bus to be early. From \( t = 0 \) onwards, create a peak around 5 minutes, and taper it off after 5 minutes to represent that it's less likely to be significantly late.

Step 3: Shading the Probability Region

Shade the area under the curve of \( p(t) \) between \( t = 2 \) and \( t = 4 \), which represents the probability that the bus is late by between 2 and 4 minutes. This area is the integral of \( p(t) \) from 2 to 4 minutes.

Step 4: Understanding the Cumulative Distribution Function

A cumulative distribution function \( P(t) \) represents the probability that a random variable is less than or equal to \( t \). It is a non-decreasing function starting at 0 and approaching 1 as \( t \) increases.

Step 5: Sketching the Cumulative Distribution Function

On a separate graph, draw the horizontal axis for time \( t \) and a vertical axis for probability. Start \( P(t) \) at 0 when \( t = -2 \) (the bus never more than 2 minutes early), increasing gradually towards 1 as \( t \) passes 5 minutes. The sharpest increase should correlate with the peak of \( p(t) \).

Step 6: Identifying Correspondences and Inflection Points

The area identified between \( t = 2 \) and \( t = 4 \) on the \( p(t) \) graph corresponds to the difference \( P(4) - P(2) \) on the \( P(t) \) graph. Inflection points on \( P(t) \) (where the curve changes concavity) correspond to peaks or troughs in \( p(t) \). These points represent the transition phases in probability density; on \( P(t) \), it interprets changes in the rate of increasing likelihood.

Key Concepts

Understanding the Cumulative Distribution Function

A cumulative distribution function (CDF), denoted as \( P(t) \), is a critical concept in probability and statistics. It helps to understand how the probability of a random event accumulates as we move through different values of \( t \). Essentially, for any given value, \( t \), on the graph, \( P(t) \) tells us the probability that a random variable (e.g., the bus's lateness) will be less than or equal to \( t \). This makes the CDF a non-decreasing function—think of it as a staircase that never goes down, only up or sideways.
To sketch a CDF, you start with zero at the lowest possible value (the rare event of the bus being up to 2 minutes early), and aim to reach one at the upper bound of the lateness timespan. Between 0 minutes (on-time) and the peak lateness, the slope on the CDF graph mirrors the density of the values. Where you observe the steepest climb on the CDF, this corresponds to the region within the PDF where the bus's lateness is most probable. As the bus never reaches extreme lateness beyond 5 minutes, the graph gradually levels out as it approaches 1, indicating a diminishing chance that the bus will be later than this.
The area between two points on this CDF graph, such as between \( t = 2 \) and \( t = 4 \), can be interpreted as the difference \( P(4) - P(2) \). This difference directly translates to the probability that the bus's delay falls between these two time marks.

Insight into Probability Distribution

Probability distribution, particularly the probability density function (PDF), is a concept used to describe the likelihood of all possible outcomes in a given context. For our bus lateness example, the PDF, \( p(t) \), specifically models how likely it is for the bus to be early, on time, or late by different amounts of time.
In constructing \( p(t) \), consider that the bus is mostly late but not excessively; thus, the function will be skewed towards positive \( t \). The horizontal axis represents time in minutes, and the vertical axis denotes the probability density. The peak of the PDF occurs where the bus is observed to be most frequently late—around 5 minutes. Following the peak, the density decreases, highlighting that significant delays are infrequent.
It is important to note that the total area under the curve of the PDF from start to end must equal 1, representing 100% probability distributed across all time values. When you shade the area between specific time bounds, like between 2 and 4 minutes, this highlights the probability of the bus being late within that interval. The integral of the PDF over any range gives the probability of an outcome residing within that range.

Examining Inflection Point Analysis

Inflection points are a core concept to grasp when analyzing cumulative distribution functions. An inflection point on the graph of a CDF is a spot where the curve changes direction in terms of concavity. In simpler terms, it’s where the graph transitions from curving upwards to downwards or vice versa.
In the context of our bus lateness example, these points are important as they pinpoint moments of significant change in the density of the lateness times. They directly correspond to peaks or valleys in the probability density function (PDF). If the PDF has a peak at a certain time, indicating that the bus is most likely late by that amount of time, this will be reflected as an inflection point on the CDF, marking where the rate of cumulative probability is increasing fastest.
Interpreting these points without referring to the PDF involves observing how quickly probabilities accumulate over small time intervals. An inflection point reflects a pivotal transition in the likelihood's growth, echoing a shift in real-world occurrences—like the bus shifting from being regularly late to only occasionally late beyond a certain time frame.
Understanding these shifts helps determine how random variables behave and ensures more accurate predictions can be made about their future behavior.