# Introduction to turbulence/Statistical analysis/Multivariate random variables

### From CFD-Wiki

### 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 positive value of indicates that and tend to vary together. A negative value indicates value indicates that when one variable is increasing the other tends to be decreasing. A zero value of indicates that there is no correlation between and . As will be seen below, it does *not* mean that they are statistically independent.

It is sometimes more convinient to deal with values of the cross-variances which have ben normalized by the appropriate variances. Thus the *correlation coefficient* is defined as:

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

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