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6 Independence in a Bivariate Distribution

6.1. Introduction

We can recognize at least three possible responses to the question, How
is Y related to X in a bivariate population?: the conditional pdf's (or
pmf's) g 2 (y∣x), the conditional expectation function E(y∣x), and the best
linear predictor E*(Y∣X). Correspondingly, we can recognize three pos-
sible responses to the question, What does it mean to say that Y is not
related to X in the bivariate population?


6.2. Stochastic Independence

In any bivariate probability distribution we can write the joint pdf (or
pmf) as the product of a conditional and a marginal pdf (or pmf):

f(x, y) = g 2 (y∣x)f 1 (x) for all (x, y) such that f 1 (x) ≠ 0.

To start, we say that Y is stochastically independent of X iff

click to view math equation
,

where

click to view math equation
does not depend on x. In other words, the conditional
probability distribution of Y given x is the same for all x-values; that is,
the conditional probability distribution does not vary with--"is indepen-
dent of"--X. One implication of stochastic independence is immediate:
the marginal pdf of Y is

click to view math equation
.

-58-

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Publication Information: Book Title: A Course in Econometrics. Contributors: Arthur S. Goldberger - author. Publisher: Harvard University Press. Place of Publication: Cambridge, MA. Publication Year: 1991. Page Number: 58.
    
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