# Introduction to turbulence/Statistical analysis/Multivariate random variables

### From CFD-Wiki

(→Joint pdfs and joint moments) |
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</td><td width="5%">(2)</td></tr></table> | </td><td width="5%">(2)</td></tr></table> | ||

- | where as before capital letters are usd to represent the mean values. Clearly the fluctuating quantities <math>u | + | where as before capital letters are usd to represent the mean values. Clearly the fluctuating quantities <math>u'</math> and <math>v'</math> are random variables with zero man. |

A psitive value of <math>\left\langle u^{'}v^{'} \right\rangle </math> indicates that <math>u^{'}</math> and <math>v^{'}</math> tend to vary together. A negative value indicates | A psitive value of <math>\left\langle u^{'}v^{'} \right\rangle </math> indicates that <math>u^{'}</math> and <math>v^{'}</math> tend to vary together. A negative value indicates |

## Revision as of 08:15, 2 June 2006

### Joint pdfs and joint moments

Often it is importamt to consider more than one random variable at a time. For example, in turbulence the three components of the velocity vector are interralated and must be considered together. In addition to the *marginal* (or single variable) statistical moments already considered, it is necessary to consider the **joint** statistical moments.

For example if and are two random variables, there are three second-order moments which can be defined , , and . The product moment is called the *cross-correlation* or *cross-covariance*. The moments and are referred to as the *covariances*, or just simply the *variances*. Sometimes is also referred to as the *correlation*.

In a manner similar to that used to build-up the probabilility density function from its measurable counterpart, the histogram, a **joint probability density function** (or **jpdf**), , can be built-up from the *joint histogram*. Figure 2.5 illustrates several examples of jpdf's which have different cross correlations. For convenience the fluctuating variables and can be defined as

| (2) |

| (2) |

where as before capital letters are usd to represent the mean values. Clearly the fluctuating quantities and are random variables with zero man.

A psitive value of indicates that and tend to vary together. A negative value indicates

### The bi-variate normal (or Gaussian) distribution

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