A city planner is analyzing two different datasets to inform policy decisions:

• Dataset 1: The declared annual income of all households in the city. The data reveals that the top 1% of households hold 30% of the city's total income, and the top 10% hold 75% of the total income.
• Dataset 2: The measured heights of all adult residents in the city. The data shows that most people's heights are very close to the average height, with very few individuals being exceptionally tall or exceptionally short.

Which dataset would be best modeled by a probability distribution that is highly skewed and has a 'heavy tail,' meaning that a small number of high-magnitude events account for a large portion of the distribution?

Analyzing Wealth Inequality with Distribution Parameters

An economist is analyzing two different datasets: one for the annual income of a country's citizens and another for the heights of those same citizens. These datasets require different probability distributions for accurate modeling. Match each distribution type with its defining characteristic.

A person named Zoë has £200 to divide between herself (amount z) and another person, Yvonne (amount y). Zoë's preferences for different divisions are represented by the utility function U(z, y) = (z-100)^2 + (y-100)^2. A higher utility value indicates a more preferred outcome. Based on this function, which of the following divisions would provide Zoë with the highest level of satisfaction?

Modeling App Success

Modeling Income Distribution: Choosing the Right Tool

A company uses a model to determine the optimal wage for its employees, represented on a graph with the real wage on the vertical axis and the number of employees on the horizontal axis. The model includes two upward-sloping curves: a 'reservation wage' curve and a 'no-shirking wage' (NSW) curve, which is positioned at a higher wage level than the reservation wage curve for any given number of employees. If the company wants to hire exactly 50 employees, and at this point on the graph the reservation wage curve indicates a wage of $18/hour and the NSW curve indicates a wage of $23/hour, what is the minimum wage the company must offer to ensure all 50 employees are motivated to work diligently?

A key characteristic of a system that can be described by a Pareto distribution is that extreme outcomes are less likely to occur compared to a system described by a normal distribution.

An economist uses a power-law probability distribution to model the wealth of two countries. The distribution for Country X is characterized by a shape parameter (alpha) of 1.5, while the distribution for Country Y is characterized by a shape parameter of 2.5. A lower value for this shape parameter indicates a 'heavier tail,' signifying a greater concentration of wealth among the highest earners. Which conclusion can be drawn from this information?

Modeling Online Content Popularity