# Cumulative Distribution function in terms of Error function

• MHB
• TheFallen018
In summary, the conversation is about finding a formula for the cumulative distribution function in terms of the error function when x is greater than or equal to 0. The error function is an even function, which can be used to find an equivalent integral form. By making a suitable substitution, the exponential in the error function can be matched with the cumulative distribution function. Algebra can then be used to combine the answers from the previous steps and obtain an expression for the distribution in terms of the error function. Further clarification can be requested if needed.
TheFallen018
Hey guys,

I've got this problem I've been trying to solve, but it makes little sense to me. I've tried a few things, but I feel like with each method, I've made no progress, and I haven't been able to make the problem make any more sense to me by trying those things.

Here's the question:

The error function is \begin{align*}\frac{1}{\sqrt{\pi}}\int_{-x}^{x}e^{-t^2} dt\end{align*}
The cumulative distribution function for $x\geq0$ is \begin{align*}\frac{1}{2}+\frac{1}{\sqrt{2\pi}}\int_{0}^{x}e^-\frac{t^2}{2} dt\end{align*}

By making a suitable substitution, find a formula for the cumulative distribution function in terms of the error function when $x\geq0$

The previous question had to do with taking the derivative with respect to x of the error function, so I was thinking I could try integrating that again and use parts of that, but that obviously didn't work. I've lost track of where my thoughts are, so I was hoping someone would be able to point me in the right direction.

Thank you :)

Last edited:
Hi TheFallen018,

Here are a few hints:
1. Notice that the error function is an even function. Use this fact to find an equivalent integral form for the error function.
2. Make a substitution in your cumulative distribution function so that $e^{-\frac{t^{2}}{2}}$ becomes $e^{-t^{2}}$ to match the form of the exponential in the error function.
3. Do a little algebra to combine your answers from Steps 1 & 2 to get an expression for the distribution in terms of the error function.

Let me know if anything requires further clarification.

Last edited:

## What is a Cumulative Distribution Function?

A Cumulative Distribution Function (CDF) is a mathematical function that represents the probability of a random variable being less than or equal to a certain value. It is commonly used in statistics to describe the distribution of a set of data.

## How is the Error Function related to the CDF?

The Error Function, denoted by erf(x), is closely related to the CDF. In fact, the CDF of a normal distribution can be expressed in terms of the Error Function. This relationship allows for the calculation of probabilities and percentiles for normal distributions using the Error Function.

## What is the significance of the CDF in statistics?

The CDF is an important concept in statistics as it allows us to calculate the probability of a random variable falling within a certain range. It also helps in visualizing the distribution of a set of data and can be used to make inferences about the population from which the data was collected.

## How is the CDF used in hypothesis testing?

In hypothesis testing, the CDF is used to calculate p-values. A p-value is the probability of obtaining a result as extreme as the observed result, assuming the null hypothesis is true. The CDF is used to calculate this probability, and if it is lower than a predetermined significance level, the null hypothesis is rejected.

## Can the CDF be used for non-normal distributions?

Yes, the CDF can be used for any type of distribution. It is not limited to only normal distributions. However, the calculation of probabilities and percentiles may be more complex for non-normal distributions compared to normal distributions.

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