Limit Of Normal CDF

by ADMIN 20 views

Introduction

The normal cumulative distribution function (CDF), denoted by Φ\Phi, is a fundamental concept in probability theory. It is defined as the integral of the standard normal probability density function (PDF) from negative infinity to a given point xx. The normal CDF has numerous applications in statistics, engineering, and other fields, including hypothesis testing, confidence intervals, and regression analysis. In this article, we will delve into the limit of the normal CDF, specifically the behavior of Φ(2log⁡(n))n\Phi\left(\sqrt{2\log(n)}\right)^n as nn approaches infinity.

The Standard Normal CDF

The standard normal CDF, denoted by Φ\Phi, is given by:

Φ(x)=12π∫−∞xe−t22dt\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-\frac{t^2}{2}} dt

This function is symmetric around zero, and its value at zero is equal to 0.5. The standard normal CDF is a monotonically increasing function, meaning that as xx increases, Φ(x)\Phi(x) also increases.

The Limit of the Normal CDF

We are interested in the limit of Φ(2log⁡(n))n\Phi\left(\sqrt{2\log(n)}\right)^n as nn approaches infinity. To approach this problem, we can use the complementary error function (erfc), which is defined as:

erfc(x)=1−Φ(x)\text{erfc}(x) = 1 - \Phi(x)

Using the erfc function, we can rewrite the expression as:

Φ(2log⁡(n))n=(1−erfc(2log⁡(n)))n\Phi\left(\sqrt{2\log(n)}\right)^n = \left(1 - \text{erfc}\left(\sqrt{2\log(n)}\right)\right)^n

The Complementary Error Function

The complementary error function (erfc) is a special function that arises in the study of probability distributions. It is defined as:

erfc(x)=2π∫x∞e−t2dt\text{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt

The erfc function is a monotonically decreasing function, meaning that as xx increases, erfc(x)(x) decreases.

The Limit of the Complementary Error Function

We can use the following inequality to bound the erfc function:

\text{erfc}(x) \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}}<br/> **The Limit of the Normal CDF: A Probabilistic Enigma** =========================================================== **Q&A: The Limit of the Normal CDF** ----------------------------------- **Q: What is the normal cumulative distribution function (CDF)?** --------------------------------------------------------- A: The normal CDF, denoted by $\Phi$, is a fundamental concept in probability theory. It is defined as the integral of the standard normal probability density function (PDF) from negative infinity to a given point $x$. **Q: What is the standard normal CDF?** -------------------------------------- A: The standard normal CDF, denoted by $\Phi$, is given by: $\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-\frac{t^2}{2}} dt

Q: What is the complementary error function (erfc)?

A: The complementary error function (erfc) is a special function that arises in the study of probability distributions. It is defined as:

erfc(x)=2π∫x∞e−t2dt\text{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt

Q: How is the erfc function related to the normal CDF?

A: The erfc function is related to the normal CDF by the following inequality:

\text{erfc}(x) \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \leq \frac{2}{