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Introduction to turbulence/Statistical analysis/Multivariate random variables

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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
v' = v - V

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

A positive value of \left\langle u'v' \right\rangle indicates that u' and v' 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 \left\langle u'v' \right\rangle indicates that there is no correlation between u' and v'. 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:

\rho_{uv}\equiv \frac{ \left\langle  u'v' \right\rangle}{ \left[ \left\langle u'^{2} \right\rangle \left\langle  v'^{2} \right\rangle \right]^{1/2}}

The correlation coefficient is bounded by plus or minus one, the former representing perfect correlation and the latter perfect anti-correlation.

As with the single-variable pdf, there are certain conditions the joint probability density function must satisfy. If B_{uv}\left( c_{1}c_{2} \right) indicates the jpdf of the random variables u and v, then:

  • Property 1
B_{uv}\left( c_{1}c_{2} \right) > 0


  • Property 2
Prob \left\{ c_{1} < u < c_{1} + dc_{1} , c_{2} < v < c_{2} + dc_{2} \right\} = B_{uv}\left( c_{1}c_{2} \right) dc_{1}, dc_{2}
  • Property 3
\int^{\infty}_{ - \infty} \int^{\infty}_{ - \infty}  B_{uv}\left( c_{1}c_{2} \right) dc_{1} dc_{2} = 1
  • Property 4
\int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{2} = B_{u}\left( c_{1} \right)

where B_{u} is a function of c_{1} only

  • Property 5
\int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{1} = B_{v}\left( c_{2} \right)

where B_{v} is a function of c_{2} only

The functions B_{u} and B_{v} are called the marginal probability density functions and they are simply the single variable pdf's defined earlier. The subscript is used to indicate which variable is left after the others are integrated out. Note that  B_{u}\left( c_{1} \right) is not the same as  B_{uv}\left( c_{1},0 \right) .

The bi-variate normal (or Gaussian) distribution


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