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A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
- Example 1: Weather Forecasting
- Example 2: Sales Tracking
- Example 3: Health Insurance
- Example 4: Traffic
- Example 5: Investing
- Example 6: Medical Studies
- Example 7: Manufacturing
- Example 8: Urban Planning
Statistics is used heavily in the field of weather forecasting. In particular, probability is used by weather forecasters to assess how likely it is that there will be rain, snow, clouds, etc. on a given day in a certain area. Forecasters will regularly say things like “there is a 90% chance of rain today between after 5PM” to indicate that there’s...
Retail companies often use descriptive statisticslike the mean, median, mode, standard deviation, and interquartile range to track the sales behavior of certain products. This gives companies an idea of how many products they can expect to sell during different time periods and allows them to know how much they should keep in inventory.
Health insurance companies often use statistics and probability to determine how likely it is that certain individuals will spend a certain amount on healthcare each year. For example, an actuary at a health insurance company might use factors like age, existing medical conditions, current health status, etc. to determine that there’s a 80% probabi...
Traffic engineers regularly use statistics to monitor total traffic in different areas of a city, which allows them to decide whether or not they should add or remove roads to optimize traffic flow. Also, traffic engineers often use time series analysisto monitor how traffic changes throughout the day so they can optimize the behavior of traffic li...
Investors use statistics and probability to assess how likely it is that a certain investment will pay off. For example, a given investor might determine that there is a 5% chance that the stock of company A will increase 100x during the upcoming year. Based on this probability, they’ll decide how much of their portfolio to invest in the stock.
Statistics is regularly used in medical studies to understand how different factors are related. For example, medical professions often use correlationto analyze how factors like weight, height, smoking habits, exercise habits, and diet are related. If a certain diet and overall weight is found to be negatively correlated, a medical professional ma...
Statistics is often used in manufacturing to monitor the efficiency of different processes. For example, manufacturing engineers may collect a random sampleof widgets from a certain assembly line and track how many of the widgets are defective. They may then perform a one proportion z-testto determine if the proportion of widgets that are defective...
Statistics is regularly used by urban planners to decide how many apartments, shops, stores, etc. should be built in a certain area based on population growth patterns. For example, if an urban planner sees that population growth in a certain part of the city is increasing at an exponential rate compared to other parts of the city then they may dec...
A function $f(x)$ is said to be quasi-concave if its domain and all its $\alpha$-superlevel sets defined as $$\mathcal{S}_\alpha \triangleq\{x|x\in dom f, f(x)\geq\alpha\}$$ are convex for every $\alpha$.
Definition. A function is concave up if the rate of change is increasing. A function is concave down if the rate of change is decreasing. A point where a function changes from concave up to concave down or vice versa is called an inflection point.
Apr 17, 2024 · Convexity and concavity are crucial in analyzing investment returns and risk management. Convex curves often represent increasing returns, signaling a good investment under certain conditions. Concave curves, on the other hand, can indicate diminishing returns, prompting a reevaluation of strategies.
Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
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Nov 14, 2023 · A concave function (sometimes called ''concave down'') is a function for which any secant line (a line connecting two points on the function) lies below the function. A concave function...
