10182

# Rotational Symmetry of Multivariate Uncorrelated Standard Normal Distribution

Rotational symmetry is a key property of the multivariate uncorrelated standard normal distribution. The top part of this Demonstration displays the contour and the 3D plots of a bivariate uncorrelated standard normal distribution. Both plots are intersected by two vertical planes or half-spaces. The radial plane runs through the origin and the tangential plane is perpendicular to it at a variable distance. The bottom parts display the density profiles created by the intersections. These profiles are standard normals up to a normalizing constant.

### DETAILS

Notice that the density profile on the lower-left plane remains unchanged for any direction of the radial plane, whereas the profile corresponding to the tangential plane diminishes as the plane moves away from the origin. However, the latter profile is the same for a plane at a given distance from the origin.
This property is useful when performing risk quantification using reliability analysis of systems involving more than one correlated random variable. The original variates are transformed into the uncorrelated standard normal space and the probability of failure is computed with the help of the transformed space.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.