Problem 15 - Applied Calculus

Question

A congressional committee is investigating a defense contractor whose projects often incur cost overruns. The data in Table 7.7 show \(y,\) the fraction of the projects with an overrun of at most \(C \%\). (a) Plot the data with \(C\) on the horizontal axis. Is this a density function or a cumulative distribution function? Sketch a curve through these points. (b) If you think you drew a density function in part (a), sketch the corresponding cumulative distribution function on another set of axes. If you think you drew a cumulative distribution function in part (a), sketch the corresponding density function. (c) Based on the table, what is the probability that there will be a cost overrun of \(50 \%\) or more? Between \(20 \%\) and \(50 \%\) ? Near what percent is the cost overrun most likely to be? Fraction, \(y,\) of overruns that are at most \(C \%\) $$\begin{array}{c|c|c|c|c|c|c|c|c}\hline C & -20 \mathrm{s} & -10 \% & 0 \% & 10 \% & 20\mathrm{se} & 30 \mathrm{se} & 40 \mathrm{se} & 50 \mathrm{st} \\\\\hline y & 0.01 & 0.08 & 0.19 & 0.32 & 0.50 & 0.80 & 0.94 & 0.99 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The plot is a cumulative distribution function. P(overrun ≥ 50%) = 0.01, P(20% ≤ overrun < 50%) = 0.49, most likely at 30-40%.

Step 1: Understanding the Table and Data

The table shows the cumulative distribution function (CDF) data. It provides the fraction, \(y\), of projects with costs overrunning by at most \(C\%\), including values for both negative cost changes (cost savings) and cost increases.

Step 2: Plotting the Data with C on the Horizontal Axis

Plot the data points \((-20, 0.01), (-10, 0.08), (0, 0.19), ..., (50, 0.99)\) on a graph with \(C\) on the horizontal axis and \(y\) on the vertical axis. Connect the points using a curve. Since the fraction \(y\) represents the proportion of projects up to \(C\%\), this curve behaves like a cumulative distribution function (CDF), typically starting at 0 and approaching 1.

Step 3: Identifying the Type of Function

The curve plotted in Step 2 is a cumulative distribution function because it shows the cumulative probability of cost overruns by each \(C\%\), increasing from 0 to 1.

Step 4: Sketching the Corresponding Density Function

Draw the probability density function (PDF) based on the CDF. The PDF is the derivative of the CDF, representing the rate of change of the cumulative probability. It indicates how the probability density increases as \(C\) varies.

Step 5: Calculating the Probabilities from the Table

To find the probability that there is a cost overrun of 50% or more, subtract the CDF value at 50% from 1. For an overrun between 20% and 50%, subtract the CDF at 20% from the CDF at 50%. These calculations yield the probabilities for different overrun intervals.

Step 6: Finding the Most Likely Overrun Percentage

Determine where the PDF peaks to find the most probable cost overrun percentage, as probability density is highest there.

Key Concepts

Probability Density Function

The Probability Density Function (PDF) is a fundamental concept in statistics and probability theory. It describes how the probability is distributed over different possible values of a continuous random variable. In simpler terms, it tells us how likely it is for a particular value to occur within an interval. Unlike probabilities for discrete values, which are sums of probabilities, the PDF gives a probability density.
This distinction is crucial because, in continuous spaces, the probability of any single point is technically zero. Instead, the probability that a value falls within a certain range is found by integrating or finding the area under the PDF curve over that range.
Often, the PDF is derived from the cumulative distribution function (CDF), which represents the cumulative probability up to a certain point. For example, if you're looking at cost overruns like in the exercise, you can see how frequently certain levels of overruns occur by examining the PDF. Imagine the plotted CDF, then the PDF would be its slope at any given point, showing how quickly the cumulative probability is changing.

• The PDF helps identify the most likely value or range and where the density is higher.
• This can indicate where the cost overruns are concentrated or most probable.
• The peak of the PDF shows the most likely or frequent outcome in your data set.

Cost Overrun Analysis

Cost overrun analysis is a method used to assess the extent to which a project's costs exceed its budgeted or expected expenditures. It is an important concept in project management, particularly in sectors like construction, defense, and IT, where budgets can often balloon unexpectedly.
The exercise provided involves understanding the extent of cost overruns in defense contracts, using data to evaluate the empirical cumulative distribution function for these overruns.
This analysis involves plotting data which reflects the fractions of projects with cost overruns up to certain percentages. It essentially reveals how frequent different levels of overruns are. This assists stakeholders in identifying the expected range of overruns and preparing better for future budgets.

• By analyzing the CDF, you get a visual representation of cumulative probabilities for different overrun thresholds.
• Understanding these patterns can aid in predicting potential financial impacts and setting aside contingency reserves.
• Cost overrun analysis helps recognize trends that might need corrective actions.

Cumulative Probability

Cumulative Probability is a key concept when dealing with probability distributions, especially in continuous settings. It is represented by the cumulative distribution function (CDF), which indicates the probability that a random variable takes on a value less than or equal to a specific value.
In the context of the exercise, the CDF was used to depict the probability of cost overruns being less than or equal to particular percentages. For instance, a CDF value of 0.50 at 20% indicates that 50% of the projects incurred costs over 20%.
Using the CDF, one can easily extract useful probability information for decision-making and risk assessment.

• Cumulative probability helps to understand the likelihood of different scenarios within a cost overrun context.
• It provides a comprehensive view of how likely it is for projects to adhere to budget limits.
• The CDF values can also be used to determine probabilities for ranges, identifying sections of particular interest or concern.

Understanding cumulative probability in such scenarios aids in making data-driven decisions, better resource allocation, and improved risk management strategies.