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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:

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:

and analogously for . In general
E 
(4.55) 
 Variance:

and analogously for .
 Covariance:

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

 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. ):
Figure:
Example of bivariate normal distribution.

Marginal distributions:
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
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Giulio D'Agostini
20030515