A heavy-tailed distribution is a type of probability distribution that has a tail, which is the part of the distribution that represents extreme values, that is significantly heavier or more significant than that of the exponential distribution. This means that it has a higher probability of producing values far from the mean compared to lighter-tailed distributions, such as the normal distribution. In practical terms, this implies that heavy-tailed distributions can model phenomena where extreme events have a considerable chance of occurring.

The International Congress of Actuaries (ICA) is a significant global event for professionals in the actuarial field, organized to address advancements, challenges, and innovations in actuarial science, insurance, pensions, and risk management. It typically brings together actuaries and experts from around the world to exchange knowledge, share research, and discuss the latest trends and developments in the industry.

The Kaplan–Meier estimator is a statistical tool used to estimate the survival function from lifetime data. It is particularly useful in medical research for analyzing time-to-event data, such as the time until an event of interest occurs (like death, relapse, or failure) when some subjects are censored, meaning they leave the study or do not experience the event during the observation period.

Life expectancy is a statistical measure that estimates the average number of years a person can expect to live, based on demographic factors such as current age and sex, as well as historical mortality rates. It is commonly used to assess the overall health and longevity of populations and can vary significantly between different countries, regions, and demographic groups due to factors like healthcare access, lifestyle, economic conditions, and environmental influences.

Longevity risk refers to the potential financial risk that arises from individuals living longer than expected. This risk is particularly relevant in contexts such as pensions, insurance, and retirement planning. Here are some key points about longevity risk: 1. **Definition**: Longevity risk is the risk that people will outlive their financial resources due to an increase in life expectancy. This can impact both individuals and financial institutions.

Mortality forecasting is the process of predicting future mortality rates within a population. This practice is vital for various fields, including public health, insurance, and demography, as it helps to estimate life expectancy, plan for healthcare needs, allocate resources, and assess the financial stability of pension and insurance systems. The purpose of mortality forecasting can include: 1. **Public Health Planning**: Governments and health organizations use mortality forecasts to allocate healthcare resources and design public health programs to improve population health.

Panjer recursion is a recursive algorithm used in actuarial science and insurance mathematics to calculate the distribution of the sum of independent random variables, particularly in the context of risk management and insurance claims. Named after Hendrik Panjer, this method is particularly useful for computing the probabilities associated with different outcomes of aggregate claims. ### Key Elements of Panjer Recursion: 1. **Assumptions**: - The random variables (e.g., claims) are independent.

A 3-way lamp is a type of lamp that can produce three different levels of light intensity, typically achieved by using a 3-way light bulb and a compatible lamp socket. These lamps are commonly used for versatility in lighting, allowing users to change the brightness to suit various activities and moods, such as reading, relaxing, or creating ambiance.

The Pareto distribution is a power-law probability distribution that is used to describe phenomena where a small number of occurrences account for a large proportion of the effect. Named after the Italian economist Vilfredo Pareto, it is often used to model the distribution of wealth, resources, and other types of measurable assets.

Predictive analytics is a branch of data analytics that uses statistical algorithms, machine learning techniques, and historical data to identify the likelihood of future outcomes. Essentially, it involves analyzing current and historical data to make predictions about future events. Here are some key elements of predictive analytics: 1. **Data Collection**: Gathering relevant data from various sources, which can include structured data (like databases) and unstructured data (like social media or sensor data).

A replicating portfolio is a financial strategy that involves creating a new portfolio of assets that closely mimics the cash flows and risk profile of another asset or portfolio, often referred to as the "target" asset. This technique is commonly used in finance to replicate the performance and characteristics of a derivative, such as an option, using a combination of underlying assets, such as stocks and bonds.

RiskMetrics is a set of financial risk management tools and methodologies developed by J.P. Morgan to measure and manage market risk. It was originally introduced in the early 1990s and has since become an industry standard for quantifying risk exposures in financial portfolios.

Risk parity is an investment strategy that aims to allocate risk rather than capital in a portfolio. The central idea behind risk parity is to balance the amount of risk taken across various asset classes—such as equities, bonds, commodities, and others—rather than simply allocating funds based on expected returns or market capitalizations.

Stock sampling, more commonly referred to in the context of inventory sampling or stock inventory sampling, involves selecting a subset of items from a larger inventory to estimate or analyze certain characteristics about the entire stock without needing to inspect every item. This method is often used in quality control, auditing, or inventory management.

The Time Value of Money (TVM) is a financial principle that explains how the value of money changes over time due to factors such as interest rates and inflation. The core idea is that a specific amount of money today has a different value compared to the same amount in the future. This difference arises from the potential earning capacity of money, which can be invested to earn interest or returns over time.

A Truncated Regression model is a type of statistical model used to analyze data when the dependent variable is only observed within a certain range, meaning that observations outside this range are not included in the dataset at all. This is different from censored data, where the values outside a certain range are still present but are only partially observed. ### Key Characteristics of Truncated Regression: 1. **Truncation**: In truncated data, observations below or above certain thresholds are entirely excluded from the analysis.

Ulpian's life table, also known as the Table of Life (Tabula Vitae), is an ancient Roman text attributed to the jurist Domitius Ulpianus, who lived in the 2nd and 3rd centuries AD. Although the original table itself has not survived, it is known that Ulpian contributed significantly to the field of legal thought and population studies in ancient Rome.

Ciro Santilli tried to add this example to Wikipedia, but it was reverted, so here we are, see also: Section "Deletionism on Wikipedia".

This is a good first example of a field of a finite field of non-prime order, this one is a prime power order instead.

$4=2^2$, so one way to represent the elements of the field will be the to use the 4 polynomials of degree 1 over GF(2):

Note that we refer in this definition to anther field, but that is fine, because we only refer to fields of prime order such as GF(2), because we are dealing with prime powers only. And we have already defined fields of prime order easily previously with modular arithmetic.

Over GF(2), there is only one irreducible polynomial of degree 2:

Addition is defined element-wise with modular arithmetic modulo 2 as defined over GF(2), e.g.:

Multiplication is done modulo $X^2+X+1$, which ensures that the result is also of degree 1.

For example first we do a regular multiplication:

Without modulo, that would not be one of the elements of the field anymore due to the $1X^2$!

So we take the modulo, we note that:and by the definition of modulo:which is the final result of the multiplication.

TODO show how taking a reducible polynomial for modulo fails. Presumably it is for a similar reason to why things fail for the prime case.

Value at Risk (VaR) is a financial metric used to assess the potential loss in value of an asset or portfolio over a defined period for a given confidence interval. It is commonly employed in the fields of risk management, investment analysis, and regulatory compliance. VaR provides a way to quantify the level of financial risk within a firm or portfolio over a specific time frame.

A Vector Generalized Linear Model (VGLM) is an extension of Generalized Linear Models (GLMs) that allows for modeling multivariate responses. In traditional GLMs, we model a single response variable contingent on predictors using a link function and an appropriate distribution from the exponential family. In contrast, VGLMs handle multiple response variables that may be correlated or influenced by the same set of predictors.