# Introduction to turbulence/Statistical analysis/Multivariate random variables

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 Revision as of 08:15, 2 June 2006 (view source)Michail (Talk | contribs) (→Joint pdfs and joint moments)← Older edit Revision as of 08:15, 2 June 2006 (view source)Michail (Talk | contribs) (→Joint pdfs and joint moments)Newer edit → Line 19: Line 19:
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- where as before capital letters are usd to represent the mean values. Clearly the fluctuating quantities $u^{'}$ and $v^{'}$ are random variables with zero man. + where as before capital letters are usd to represent the mean values. Clearly the fluctuating quantities $u'$ and $v'$ are random variables with zero man. A psitive value of $\left\langle u^{'}v^{'} \right\rangle$  indicates that $u^{'}$ and $v^{'}$ tend to vary together. A negative value indicates A psitive value of $\left\langle u^{'}v^{'} \right\rangle$  indicates that $u^{'}$ and $v^{'}$ 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 $u$ and $v$ are two random variables, there are three second-order moments which can be defined $\left\langle u^{2} \right\rangle$ , $\left\langle v^{2} \right\rangle$ , and $\left\langle uv \right\rangle$. The product moment $\left\langle uv \right\rangle$ is called the cross-correlation or cross-covariance. The moments $\left\langle u^{2} \right\rangle$ and $\left\langle v^{2} \right\rangle$ are referred to as the covariances, or just simply the variances. Sometimes $\left\langle uv \right\rangle$ 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),$B_{uv}$ , 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 $u^{'}$ and $v^{'}$ can be defined as

 $u^{'} = u - U$ (2)
 $v^{'} = v - V$ (2)

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

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

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