Question

For the proof of Theorem 5.8.1, we assumed that the cdf was strictly increasing over its support. Consider a random variable \(X\) with cdf \(F(x)\) which is not strictly increasing. Define as the inverse of \(F(x)\) the function $$ F^{-1}(u)=\inf \\{x: F(x) \geq u\\}, \quad 0

Short answer

The function \(F^{-1}(u)=\inf \{x: F(x) \geq u\}\) is the generalized inverse of the cdf \(F(x)\). It signifies the smallest value such that \(F(x)\) is greater than or equal to \(u\). However, when \(F\) is not strictly increasing, \(F^{-1}(u)\) might not return a unique value and could instead return a range of values.

Step-by-step solution

Clarify the Definitions

First, it's key to understand what each part of the equation \(F^{-1}(u)=\inf \{x: F(x) \geq u\}\) represents: - \(F^{-1}(u)\) is the function we're trying to define, often called the 'generalized inverse' of the cdf. - The \(\inf \{x: F(x) \geq u\}\) part represents the smallest \(x\) that's still greater than or equal to \(u\). In this context, 'smallest' means the lowest value which still makes the inequality true.

Understand the meaning of the inverse cdf

The generalized inverse cdf \(F^{-1}(u)\) is defined as the smallest value in the support of \(X\) that still gives a probability equal to or larger than \(u\). When \(F\) is not strictly increasing, this value might not be unique; there may be many values in the support of \(X\) that meet this criteria. Therefore, it is crucial to understand that although \(F^{-1}(u)\) may not return a unique \(x\) value, it always picks the smallest possible one meeting the criteria.

Handling the infimum

Applying infimum might not be straightforward in most cases. For this problem, the infimum of the set \{x: F(x) \geq u\} refers to the smallest \(x\) value that makes \(F(x)\) greater than or equal to \(u\). However, it's important to note that if \(F(x)\) is not strictly increasing, there might not be a unique smallest value. Therefore, even though the infimum always exists, it might be a range of values rather than a specific number.

Key concepts

Cumulative Distribution Function

A Cumulative Distribution Function (CDF) is a fundamental concept in probability and statistics that describes the probability that a random variable takes on a value less than or equal to a certain value. It is denoted as \( F(x) \). The CDF gives you a cumulative sum of probabilities up to a point \( x \). This makes it a vital tool in understanding the behavior and distribution of random variables.

Key properties of a CDF include:

• The CDF is non-decreasing; that is, as \( x \) increases, \( F(x) \) can only remain the same or increase, never decrease.
• It ranges from 0 to 1, where \( F(x) = 0 \) as \( x \) approaches minus infinity and \( F(x) = 1 \) as \( x \) approaches infinity.
• It is right-continuous, meaning that at any point \( x \), the value of the function from the right approaches \( F(x) \).

The important takeaway is that the CDF helps in determining probabilities for different intervals and is easier to use when calculating cumulative probabilities compared to probability density functions (PDFs) or probability mass functions (PMFs).
Understanding a CDF also plays a critical role in working with its inverse, which is useful in practice for simulations and generating random variables.

Infimum

The concept of infimum is a bit abstract but crucial in mathematical analysis. The infimum of a set of numbers is the greatest number that is less than or equal to every number in the set. In simpler terms, it's like a lower boundary that a set of numbers approaches but may not necessarily reach.

In the context of inverse CDF: when we talk about the infimum \( \inf \{x : F(x) \geq u \} \), we are looking for the smallest \( x \) such that the CDF \( F(x) \) is greater than or equal to a particular value \( u \). This is particularly useful when the CDF is not strictly increasing.

The infimum comes into play when there might be multiple values of \( x \) satisfying \( F(x) \geq u \). For many purposes, including defining generalized inverse functions, it helps pinpoint the smallest such value. Even if there are multiple values that satisfy the constraint, the infimum serves as the most meaningful representation of that set's smallest boundary.

Generalized Inverse

A generalized inverse of a function, particularly in the context of a CDF, expands our conventional understanding of inverse functions. In typical scenarios, an inverse function \( F^{-1}(u) \) is expected to return a unique result. However, when dealing with CDFs that aren't strictly increasing, the concept of a generalized inverse becomes useful.

The generalized inverse determines the smallest \( x \) for which the CDF \( F(x) \) is at least equal to \( u \). Even if \( F(x) \) plateaus, as it can happen with non-strictly increasing functions, the generalized inverse picks out the first value that still satisfies the condition \( F(x) \geq u \).

This concept is integral in real-world applications such as inverse transform sampling, where you may need to simulate random variables from a probability distribution. Utilizing the generalized inverse ensures that when sampling values, you start with the smallest acceptable possibility given the CDF’s behavior.

• The generalized inverse answers the "what minimum value is still good enough" question when dealing with CDFs.
• It accommodates CDFs that are not perfectly aligned to have a strict one-to-one correspondence between values and their probabilities.

Understanding the generalized inverse is critical in scenarios where precision and dealing with non-uniqueness are required in probabilistic models.