Page 164 - Probability and Statistical Inference

3. Multivariate Random Variables 141

coefficient between X and X will amount to saying that p = p p where so far
1
1 2
2
p, p and p have all been assumed to lie between (0, 1) but they are otherwise
2
1
arbitrary. Now, we must have then P(X = 1∩X = 0) = p – p, and P(X =
1
1
2
1
0nX = 0) = 1–p –p +p, and P(X =0∩X =1) = p –p. But, now P(X = 0nX =
1
2
2
1
2
2
2
1
1) = p –p = p –p p = p (1–p ) = P(X =0) P(X =1); P(X = 1 ∩ X = 0) = p –
2 2 1 2 2 1 1 2 1 2 1
p = p – p p = p (1 – p ) = P(X = 1) P(X = 0); and P(X =0 ∩ X =0) = 1–p –
1 2
1
2
1
1
1
1
2
2
p +p = 1–p –p +p p = (1–p )(1–p ) = P(X =0) P(X =0). Hence, the two such
1
1 2
1
2
2
1
2
2
random variables X and X are independent. Here, the zero correlation coef-
1
2
ficient implied independence, in other words the property that “the zero cor-
relation coefficient implies independence” is not a unique characteristic prop-
erty of a bivariate normal distribution. !
Example 3.7.4 There are other simple ways to construct a pair of random
variables with the zero correlation coefficient. Start with two random vari-
ables U , U such that V(U ) = V(U ). Let us denote X = U + U and X = U 1
1
2
1
2
2
1
2
1
– U . Then, use the bilinear property of the covariance function which says
2
that the covariance function is linear in both components. This property was
stated in the Theorem 3.4.3, part (iv). Hence, Cov(X , X ) = Cov(U + U , U
1 2 1 2 1
– U ) = Cov(U , U ) – Cov(U , U ) + Cov(U , U ) – Cov(U , U ) = V(U ) –
1
2
1
2
2
2
2
1
1
1
V(U ) = 0. !
2
3.8 The Exponential Family of Distributions
The exponential family of distributions happens to be very rich when it comes
to statistical modeling of datasets in practice. The distributions belonging to
this family enjoy many interesting properties which often attract investigators
toward specific members of this family in order to pursue statistical studies.
Some of those properties and underlying data reduction principles, such as
sufficiency or minimal sufficiency, would impact significantly in Chapter 6
and others. To get an idea, one may simply glance at the broad ranging results
stated as Theorems 6.2.2, 6.3.3 and 6.3.4 in Chapter 6. In this section, we
discuss briefly both the one-parameter and multi-parameter exponential fami-
lies of distributions.
3.8.1 One-parameter Situation
Let X be a random variable with the pmf or pdf given by f(x; θ), x ∈ χ ⊆ ℜ,
θ ∈ Θ ⊆ ℜ. Here, θ is the single parameter involved in the expression of f(x;
θ) which is frequently referred to as a statistical model.