Tagged Questions
1
vote
1answer
101 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 ...
4
votes
1answer
99 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 ...
0
votes
0answers
101 views
Integration involving product of inverse error and polynomial functions
I am stuck with the following integral:
$$\int_{x \ge 1} \frac{18 \exp \left(-\frac12 \left(\phi^{-1}[1 - \frac{1}{x^3}]\phi^{-1}[1 - \frac{1}{(z-x)^3}] \right) \right)}{\sqrt{3}x^3 (z-x)^4} dx $$
...
1
vote
0answers
197 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?
$$ ...
27
votes
2answers
2k views
Why is the error function defined as it is?
$\newcommand{\erf}{\operatorname{erf}}$
This may be a very naïve question, but here goes.
The error function $\erf$ is defined by
$$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$
Of ...
