1. Why we need Lower Partial Moments measure? How we apply this? a. Problem with Symmetric Measures: Traditional risk measures such as variance focus on deviations from the mean in both positive and negative directions. However, decision-makers are typically more concerned about losses than gains. Variance does not distinguish between "bad" (losses) and "good" (gains) deviations from expectations.
b. Focus on Downside Risk: LPM specifically focuses on downside risk, meaning only losses are considered. This aligns with the primary concern of a decision-maker. LPM provides a more general approach for introducing a critical value or threshold (z). This threshold defines the point below which an outcome is considered a "loss" or "harm" relevant to the decision-maker, such as a fixed cost level or an expected value
LPM0 (Order 0): This measures the probability that the outcome falls below the defined threshold z. For example, for an investment, it calculates the probability of obtaining a certain loss level. LPM1 (Order 1): This measures the "expected shortfall" or the average magnitude of the loss below the threshold z. Or it indicates the mean loss. This can help understand how big gap till threshold on average. LPM2 (Order 2): This measures a "shortfall variance" or the expected squared loss below the threshold z. This helps to see/capture most severe cases, outliers. 2. How identify that we might need to apply ths method for task? What is given usually? What we know, what data/conditions is given Identify the "data" (all given information, conditions, initial states, or parameters). Identify the "conditions" (rules, constraints, definitions, or allowable operations). Identify the explicit "unknown", what is question here (what needs to be found, proven, or achieved). Define Expected Outcome:
◦ Clearly articulate what a complete and correct solution looks like. Is it a specific numerical value, a proof, a set of solutions, or a general formula?
◦ Understand the properties the desired outcome should possess (e.g., must be an integer, continuous, unique, etc.). Visualize task with scheme, picture, short mathematical writing or table.
◦ Draw a figure or diagram if the problem involves geometry, physical setups, or abstract relationships.
◦ Introduce suitable notation for variables, sets, or quantities.
◦ Algebraize by translating relationships into mathematical expressions or equations where appropriate. Understand the formula, without going into much details on math or statistic terms Derive, write down formula for applying to specific task 3. Proof or Solution + Write down/Highlight Answer for task Identify with what steps task is solved/prooved. Insert in formula data from the specific task to set up a task to solve Write down solution/proof step by step in a lot of details if needed Categorize the Problem: Determine the broad area(s) of mathematics or logic to which the problem belongs (e.g., Number Theory, Geometry, Combinatorics, Algebra, Functional Equations, Optimization, Proof). This helps narrow down potential strategies. Risks, safety concerns, mistakes, and how to mitigate/correct them. - Call out frequent errors and best practices for prevention or resolution 
LPM measure in assessing downside risk in cases with discrete data, Cheat Sheet