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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
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(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:
-
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|
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(4.47) |
|
|
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(4.48) |
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|
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(4.49) |
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(4.50) |
- Independent random variables
-
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(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) |
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VarCov |
(4.63) |
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(4.64) |
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(4.65) |
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(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
Contents
Giulio D'Agostini
2003-05-15