Page 142 - Probability and Statistical Inference

3. Multivariate Random Variables 119

2, ∫ ∫ ∫ ∫ ∫ dx = 2, and ∫ ∫ ∫ ∫ ∫ a (x ) dx = 2.
χ i χ 4 4 i
Hence, one verifies that ∫ ∫ ∫ ∫ ∫ = 1 so that (3.3.53) does
χ
provide a genuine pdf since f(x) is non-negative for all x ∈ χ. Along the lines
of the example 3.3.10, one should be able to write down (i) the marginal joint
pdf of (X , X , X ), (ii) the conditional pdf of X , X given the values of X , X ,
2
4
1
3
5
3
5
(iii) the conditional pdf of X given the values of X , X , (iv) the conditional
1
4
3
pdf of X given the values of X , X , X , among other similar expressions.
3
1
2
4
Look at the Exercises 3.3.9-3.3.10. Look at the Exercise 3.3.11 too for other
possible choices of a ’s. !
i
3.4 Covariances and Correlation Coefficients
Since the random variables X , ..., X vary together, we may look for a mea-
1
k
sure of the strength of such interdependence among these variables. The
covariance aims at capturing this sense of joint dependence between two real
valued random variables.
Definition 3.4.1 The covariance between any two discrete or continuous
random variables X and X , denoted by Cov (X , X ), is defined as
1 2 1 2
where µ = E(X ), i = 1, 2.
i i
On the rhs of (3.4.1) we note that it is the expectation of a specific func-
tion g(X , X ) = (X – µ )(X – µ ) of the two random variables X , X . By
2
1
2
1
1
2
1
2
combining the ideas from (3.2.6) and (3.3.8) we can easily rewrite (3.4.1).
Then, using the Theorem 3.3.2 we obtain
We now rewrite (3.4.2) in its customary form, namely

whether X , X are discrete or continuous random variables, provided that the
2
1
expectations E(X X ), E(X ) and E(X ) are finite. From (3.4.3) it also be-
1
2
1
2
comes clear that