Infinite Distribution

Extracted Attributes

• Field (Probability Theory)

  Probability Theory

• Context (Business Strategy)

  M&A Strategy in Entertainment Industry

• Applied by (Business Strategy)

  Ari Emanuel

• Definition (Probability Theory)

  A probability distribution that can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables

• Importance (Probability Theory)

  Identifying suitable distribution families for models, role in limit theorems

• Inspired by (Business Strategy)

  George Gilder

• Core Principle (Business Strategy)

  IP Ownership, consolidation of assets, focus on Live Events

• Introduced by (Probability Theory)

  Bruno de Finetti

• Year Introduced (Probability Theory)

  1929

• Strategic Outcome (Business Strategy)

  Formation of Endeavor and TKO

Timeline

• Bruno de Finetti introduced the concept of infinite divisibility of probability distributions. (Source: Wikipedia)

  1929

• Ari Emanuel applies the concept of 'Infinite Distribution' to his M&A strategy, leading to the formation of Endeavor through mergers and acquisitions including William Morris Endeavor (WME), IMG, and UFC. (Source: Related Documents)

  Unknown

• The creation of TKO through a merger between UFC and WWE, a deal made with Vince McMahon, as a culmination of Ari Emanuel's 'Infinite Distribution' strategy. (Source: Related Documents)

  Unknown

Infinite divisibility (probability)

In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function. More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n, there exist i.i.d. random variables Xn1, ..., Xnn whose sum Sn = Xn1 + ... + Xnn has the same distribution F. The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for certain models or applications. Infinitely divisible distributions play an important role in probability theory in the context of limit theorems.

Web Search Results

• Infinitely Divisible Distributions - Random Services

  Infinitely divisible distributions form an important class of distributions on \( \R \) that includes the stable distributions, the compound Poisson distributions, as well as several of the most important special parametric families of distribtions. Basically, the distribution of a real-valued random variable is infinitely divisible if for each \( n \in \N\_+ \), the variable can be decomposed into the sum of \( n \) independent copies of another variable. Here is the precise definition. [...] The distribution of a real-valued random variable \( X \) is infinitely divisible if for every \( n \in \N\_+ \), there exists a sequence of independent, identically distributed variables \( (X\_1, X\_2, \ldots, X\_n) \) such that \( X\_1 + X\_2 + \cdots + X\_n \) has the same distribution as \( X \). If the distribution of \( X \) is stable then the distribution is infinitely divisible. [...] has the same distribution as \( X \). But \( \left(\frac{X\_i - a\_n/n}{b\_n}: i \in \{1, 2, \ldots, n\} \right) \) is an IID sequence, and hence the distribution of \( X \) is infinitely divisible.

• 5.4: Infinitely Divisible Distributions - Statistics LibreTexts

  Infinitely divisible distributions form an important class of distributions on \( \R \) that includes the stable distributions, the compound Poisson distributions, as well as several of the most important special parametric families of distribtions. Basically, the distribution of a real-valued random variable is infinitely divisible if for each \( n \in \N\_+ \), the variable can be decomposed into the sum of \( n \) independent copies of another variable. Here is the precise definition. [...] The distribution of a real-valued random variable \( X \) is infinitely divisible if for every \( n \in \N\_+ \), there exists a sequence of independent, identically distributed variables \( (X\_1, X\_2, \ldots, X\_n) \) such that \( X\_1 + X\_2 + \cdots + X\_n \) has the same distribution as \( X \). If the distribution of \( X \) is stable then the distribution is infinitely divisible. [...] - a\_n/n}{b\_n}: i \in \{1, 2, \ldots, n\} \right) \) is an IID sequence, and hence the distribution of \( X \) is infinitely divisible.

• Infinite Divisibility in Probability: Definition - Statistics How To

  The Poisson distribution is an infinitely divisible distribution. For each μ >0, and each n∈ {1, 2, 3,…}, independent random variables Xn1, Xn2,…Xnn have a Poisson distribution. The normal distribution, the chi-squared distribution and the Cauchy distribution are also examples of infinitely divisible distributions. In fact, every stable distribution is infinitely divisible. However, the reverse isn’t true— not every infinitely divisible distribution is stable. [...] > A probability distribution is infinitely divisible if it can be written as the sum of n independent and identically distributed random variables for any positive integer n. Infinite divisibility is an important property of stable probability distributions. They are often seen in practical use such as waiting time theory and modeling problems as well as theoretical applications like limit theorems. ## Defining Infinite Divisibility [...] If the probability distribution F is infinitely divisible, then for any positive integer n, there is a variable Sn which can be written as Sn = Xn1 + Xn2 + …. + Xnn such that each of the Xn are random variables, independent and identically distributed; their sum, Sn, has the probability distribution F. ## Infinite Divisibility and Characteristic Functions

• [PDF] Building Infinite Processes from Finite-Dimensional Distributions

  2.1 Finite-Dimensional Distributions So, we now have X, our favorite Ξ-valued stochastic process on T with paths in U. Like any other random variable, it has a probability law or distribution, which is defined over the entire set U. Generally, this is infinite-dimensional. Since it is inconvenient to specify distributions over infinite-dimensional spaces all in a block, we consider the finite-dimensional distributions. [...] The next theorem says that this worry is unfounded; the finite-dimensional dis-tributions specify the infinite-dimensional distribution (pretty much) uniquely. Theorem 23 (Finite-dimensional distributions determine process dis-tributions) Let X and Y be two Ξ-valued processes on T with paths in U. Then X and Y have the same distribution iffall their finite-dimensional distributions agree. [...] Chapter 2 Building Infinite Processes from Finite-Dimensional Distributions Section 2.1 introduces the finite-dimensional distributions of a stochastic process, and shows how they determine its infinite-dimensional distribution.

• 7 Types of Statistical Distributions with Practical Examples

  Similarly, we can see examples of infinite outcomes from discrete events in our daily environment. Recording time or measuring a person’s height has infinitely many values within a given interval. This type of data is called Continuous Data, which can have any value within a given range. That range can be finite or infinite. [...] Depending on the type of data we use, we have grouped distributions into two categories, discrete distributions for discrete data (finite outcomes) and continuous distributions for continuous data (infinite outcomes). Dig deeper into the Discrete vs Continuous data distributions ### Discrete Distributions #### Discrete Uniform Distribution: All Outcomes are Equally Likely [...] Before we proceed further, if you want to learn more about probability distribution, watch this video below: ## Common Types of Data Explaining various distributions becomes more manageable if we are familiar with the type of data they use. We encounter two different outcomes in day-to-day experiments: finite and infinite outcomes.