Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 123

Effects on the variance of translation and change of scale:
for any real c,

VX ð þ c Þ ¼ VX ðÞ

and

c 2 VX VcX ðÞ ¼ :

A.5 Joint Distributions

The joint distribution of X = (X 1 , , p),

is the probability distribution p X on (R n ,B n ) (where B n denotes the sigma algebra
generated by the products of intervals) determined by

… ,X n ), a vector of random variables on (S, ʛ

p X I 1X ... X I n ð Þ ¼ p ð X 1 2 I 1 ; ... ; X n 2 I n Þ;

for any set of intervals (I 1 , ,I n ).

In the same way, the concepts of joint cumulative distribution function and of
joint density extend the one-dimensional case. In the context of joint distributions
of vectors X, the distribution of each one-dimensional random variable X i , is called
a marginal distribution. In the same way, its cdf is called a marginal cdf and its
density is called a marginal density.
A vector of random variables X = (X 1 ,

…

,X n ) has a continuous distribution if
and only if there is a positive function if X such that the, for F X the joint cumulative
distribution function of X,

…

x 1 x n

F X ð x 1 ; ... ; x n Þ ¼ f X x 1 ... ð x n Þ dx 1 ... dx n

...

1 1

The following concepts, of correlation and independence between random
variables, help to understand joint probability distributions.
Covariance of the pair of random variables X and Y is the expected value of the
product of their deviations to the respective means:

Cov X ð ; Y Þ ¼ EX ½ ð EX Þ ð Y EY Þ

So, if values above (or below) their expected values tend to occur together, then
X and Y have a positive covariance. If values above the expected value for one of
them tend to be accompanied by values below the expected value for the other then
they have a negative covariance.

Cov X ð ; Y Þ ¼ EXY ðÞ EX ðÞ EY ðÞ

and

ð þ Þ ¼ ðÞþ ðÞþ ð ; Þ

VX Y VX VY 2Cov X Y