0
votes
1answer
110 views

Is there a name for the normal CDF function $\Phi(\cdot)$?

I can't seem to find a plain English name for the CDF of the normal distribution $\Phi(x)$. However, I am aware of several other related functions that have a name, so I feel like this one should as ...
3
votes
2answers
54 views

Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?

Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how? This seems like an easy question, but ...
2
votes
1answer
384 views

Closed Form of Normal Distribution

What does closed form in following sentence mean and why we need tables of c.d.f.? Normal distributions's p.d.f. cannot be integrated in closed form, and hence tables of the c.d.f. or computer ...
5
votes
1answer
130 views

how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian)

I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma ...
1
vote
0answers
253 views

Nested Integral of exponential function with trigonometric identities

Is there any possibility to simplify the following integral or any function that is equivalent to the following integral? $$ ...
29
votes
2answers
4k views

Why is the error function defined as it is?

$\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of ...