Question

The error function \(y=\operatorname{erf}(x)\) is defined by the equation $$ y=\operatorname{erf}(x)=\int_{0}^{x} e^{-t^{2}} d t $$ This function is of great importance in probability and statistics and is tabulated in the books mentioned after (4.24). Establish the following properties: a) \(\operatorname{erf}(x)\) is defined and continuous for all \(x\); b) \(\operatorname{erf}(-x)=-\operatorname{erf}(x)\); c) \(-1<\operatorname{erf}(x)<1\) for all \(x\).

Short answer

The three properties of the error function, erf(x), demonstrated in the solution are: a) erf(x) is defined and continuous for all x; b) erf(-x) = -erf(x); c) -1 < erf(x) < 1 for all x.

Step-by-step solution

Property a: Defined and continuous for all x

Firstly, it is necessary to prove that erf(x) is defined and continuous for all x. The integrand \(e^{-t^2}\) is a composition of continuous functions, so it is continuous on its entire domain, \([-\infty, \infty]\). Consequently, the integral of a continuous function on a finite interval is also continuous. Therefore, erf(x) is defined and continuous for all x.

Property b: erf(-x) = -erf(x)

We will now prove that \(\operatorname{erf}(-x)=-\operatorname{erf}(x)\). The erf(-x) can be written as: $$ \operatorname{erf}(-x)=\int_{0}^{-x} e^{-t^{2}} d t $$ By changing the variable using the substitution \(u=-t\), the integral becomes: $$ \operatorname{erf}(-x)=\int_{0}^{-x} e^{u^2}(-du) $$ Now, reversing the order of the integration, we have: $$ \operatorname{erf}(-x)=-\int_{-x}^{0} e^{u^2} du $$ Finally, using the substitution \(v=-u\), we obtain: $$ \operatorname{erf}(-x)=-\int_{x}^{0} e^{-v^{2}} dv =-\operatorname{erf}(x) $$ Thus, \(\operatorname{erf}(-x)=-\operatorname{erf}(x)\).

Property c: -1 < erf(x) < 1 for all x

Lastly, let us prove that -1 < erf(x) < 1 for all x. We know that the function \(e^{-t^2}\) is always positive for all t. This means that the integral value will always be positive for a positive integrator. So, for positive x values, we have: $$ 0 < \int_{0}^{x} e^{-t^{2}} d t < \int_{0}^{x} 1 d t = x $$ As x goes to infinity, the value of the function goes to 1 because the integrand is always less than 1. Therefore, we have: $$ 0 < \operatorname{erf}(x) < 1 \textrm{ for all } x>0 $$ Likewise, from Property b, we have that \(\operatorname{erf}(-x)=-\operatorname{erf}(x)\). Thus, $$ -1 < \operatorname{erf}(-x) < 0 \textrm{ for all } x<0 $$ Combining both cases, we can conclude that -1 < erf(x) < 1 for all x.

Key concepts

Integral Properties

The study of integrals is essential not just for mathematics but for many applications, including error function analyses in probability and statistics. When discussing integral properties, we focus on aspects like continuity and the behaviors of integrals under different operations.

For example, the integral of a continuous function, such as the term \(e^{-t^2}\), is itself a continuous function when the limits of integration are finite. This fundamental property reassures us that functions defined by integrals, like the error function, maintain continuity within their domain. Furthermore, integral properties also include symmetry, which is a concept that arises when evaluating integrals with variable limits.

Continuity of Functions

Continuity is a cornerstone concept in calculus, indicating that a function has no breaks, jumps, or holes in its graph. It is required for functions like \(\operatorname{erf}(x)\) to ensure that they can be used reliably in applications such as probability models.

The essence of continuity for any function, including the error function, is that small changes in the input \(x\) produce small changes in the output \(y\), without any unexpected spikes. This is crucial for calculating probabilities, where the outcome needs to be smoothly adjusted to varying input data.

Functions in Probability

In the context of probability, functions often depict distributions and error rates. The error function, specifically, is used to describe the accuracy of predictions, quantify errors in hypothesis testing, and it's integral to the normal distribution curve. This function is bounded, as implied by the inequality \( -1 < \operatorname{erf}(x) < 1 \) for all \( x \), which ensures that probability values are kept within the [0,1] interval, a fundamental requirement in probability theory.

Understanding the behavior of such functions is pivotal in predicting outcomes and quantifying the uncertainty associated with those predictions in various fields such as statistics, finance, and engineering.

Variable Substitution

Variable substitution is a powerful tool in integral calculus, used for simplifying integrals to make them more manageable. The process involves replacing a complex part of an integral with a new variable. This technique was employed in above to prove the property \(\operatorname{erf}(-x)=-\operatorname{erf}(x)\).

By changing the variable from \( t \) to \( u \) and then to \( v \) in the integral defining the error function, we're able to manipulate the bounds and integrands into a familiar form. Through substitution, the integral's symmetry becomes apparent, aiding in proving the function's reflective properties which are pivotal in applications that require the analysis of data distributions that are symmetric about the mean.