# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# Laplace distribution

In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also known as the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together back-to-back. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution.

## Distribution, density, and quantile function

A random variable has a Laplace(μ, b) distribution if its probability density function is

$f(x) = \frac{1}{2b} \exp-\frac{|x-\mu|}{b}\,\!$
$= \frac{1}{2b} \left\{\begin{matrix} \exp-\frac{\mu-x}{b} & \mbox{if }x < \mu \\[8pt] \exp-\frac{x-\mu}{b} & \mbox{if }x \geq \mu \end{matrix}\right.$

Here, μ is a location parameter and b > 0 is a scale parameter. If μ = 0, the positive half-line is exactly an exponential distribution scaled by 1/2.

The pdf of the Laplace distribution is also reminiscent of the normal distribution; however, whereas the normal distribution is expressed in terms of the squared difference from the mean μ, the Laplace density is expressed in terms of the absolute difference from the mean. Consequently the Laplace distribution has fatter tails than the normal distribution.

The Laplace distribution is easy to integrate, if one distinguishes two symmetric cases, due to the use of the absolute value function. Its cumulative distribution function is as follows:

 F(x) $= \int_0^x \!\!f(u)\,\mathrm{d}u$ $= \left\{\begin{matrix} &\frac12 \exp-\frac{\mu-x}{b} & \mbox{if }x < \mu \\[8pt] 1-\!\!\!\!&\frac12 \exp-\frac{x-\mu}{b} & \mbox{if }x \geq \mu \end{matrix}\right.$ $=0.5\,[1 + \sgn(x-\mu)\,(1-\exp(-|x-\mu|/b))]$

The inverse cumulative distribution function is given by

$F^{-1}(p) = \mu - b\,\sgn(p-0.5)\,\ln(1 - 2|p-0.5|)$

## Variates

A Laplace(0, b) variate can be generated as the difference of two i.i.d. Exponential(1/b) variates. Equivalently, a Laplace(0, 1) variate can be generated as the logarithm of the ratio of two iid uniform variates.

Last updated: 05-31-2005 21:53:46
03-10-2013 05:06:04
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
Science kits, science lessons, science toys, maths toys, hobby kits, science games and books - these are some of many products that can help give your kid an edge in their science fair projects, and develop a tremendous interest in the study of science. When shopping for a science kit or other supplies, make sure that you carefully review the features and quality of the products. Compare prices by going to several online stores. Read product reviews online or refer to magazines.

Start by looking for your science kit review or science toy review. Compare prices but remember, Price \$ is not everything. Quality does matter.
 Science Fair Coach What do science fair judges look out for? ScienceHound Science Fair Projects for students of all ages
 All Science Fair Projects.com Site All Science Fair Projects Homepage Search | Browse | Links | From-our-Editor | Books | Help | Contact | Privacy | Disclaimer | Copyright Notice