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## Distribution of several random variables

We only consider the case of two continuous variables ( and ). The extension to more variables is straightforward. The infinitesimal element of probability is , and the probability density function

 (4.43)

The probability of finding the variable inside a certain area is

 (4.44)

Marginal distributions:

 (4.45) (4.46)

The subscripts and indicate that and are only functions of and , respectively (to avoid fooling around with different symbols to indicate the generic function), but in most cases we will drop the subscripts if the context helps in resolving ambiguities.
Conditional distributions:

 (4.47) (4.48) (4.49) (4.50)

Independent random variables

 (4.51)

(it implies and .)
Bayes' theorem for continuous random variables

 (4.52)

(Note added: see proof in section .)
Expectation value:

 E (4.53) (4.54)

and analogously for . In general

 E (4.55)

Variance:

 EE (4.56)

and analogously for .
Covariance:

 Cov E (4.57) EE   E (4.58)

If and are independent, then    EE   E and hence Cov (the opposite is true only if , ).
Correlation coefficient:

 (4.59) (4.60)

Linear combinations of random variables:

If , with real, then:
 E E (4.61) Var VarCov (4.62) VarCov (4.63) (4.64) (4.65) (4.66)

has been written in different ways, with increasing levels of compactness, that can be found in the literature. In particular, () and () use the notations    Cov and , and the fact that, by definition, .
Bivariate normal distribution:

Joint probability density function of and with correlation coefficient (see Fig. ):

 (4.67)

Marginal distributions:
 (4.68) (4.69)

Conditional distribution:

 (4.70)

i.e.

 (4.71)

The condition squeezes the standard deviation and shifts the mean of .

Next: Central limit theorem Up: Random variables Previous: Continuous variables: probability and   Contents
Giulio D'Agostini 2003-05-15