Multivariable joint probability density

The variables that govern turbulent reacting flows have large, random fluctuations. This suggests a statistical treatment of the variables which leads to the use of multivariable joint probability density functions (PDF) t). i) denotes the probability that a system at location x and time r is in a state between and + d, where denotes a particular state in the underlying sample space Q. In this notation, for instance, = (p,p, T, u, Ti,..., Yn) denotes a vector whose components are flow and thermo-chemical random variables. More precisely, p denotes the density, p the pressure, T the temperature, u the velocity, and Ti,..., Tw the N species mass fractions. 

This is a multivariable joint probability density function and therefore ... 

Hence the probability cannot be negative, and the probability that the value of X is found somewhere in U equals unity. Moreover, let us adopt the condition that the function /x is (not only integrable, but also) sufficiently small at infinity see (E.l.lOa) below. [On the other hand, we do not require/x to be continuous.] In this manner, the randomness of variable X is quantitatively characterized by the joint probability density f. The probability is distributed according to the integrals (E.1.2). The law (E.1.2) is also called the probability distribution of random variable X The probability (of the event) that the value of X is found in 2> is determined by a well-defined integral over 2>. For a random vector variable (with N > ), the distribution is called multivariate. 

The likelihood function is derived from the probability distribution of the measurement errors relating to the diagnostic functionals. An appropriate distribution is a multivariate Gaussian with independent errors. Thus the joint probability density of the diagnostics and the input is... 

---