1
$\begingroup$

If we have two Random Variables X and Y, what is the interpretation for the three cases:

a] p(X=x,Y=y) = $p_1$(X=x) * $p_2$(Y=y)

b] p(X=x,Y=y) < $p_1$(X=x) * $p_2$(Y=y)

c] p(X=x,Y=y) > $p_1$(X=x) * $p_2$(Y=y)

where p -> joint pdf of X and Y,

$p_1$ -> marginal pdf of X

$p_2$ -> marginal pdf of Y

  • 0
    Various types of correlation between X and Y2010-10-30
  • 0
    If its not in LaTeX, then it becomes tough for users to understand as to what exactly the OP Wants2010-10-30
  • 0
    @Chandru1: I apologize. I have formatted the question better. @Yonatan N: Can you please elaborate? Does a] signify X and Y are independent? What is the difference between b] and c] wrt correlations?2010-10-30
  • 0
    I would call "b" "negatively correlated". In other words, observing y decreases the probability of observing x. The reverse for c.2010-12-31
  • 0
    As pointed out by Yonatan N, these are just different types of correlations. In general, they are defined via the c.d.f. instead of the p.d.f. For example, your case (a) should be $\mathbb P(X\leq x,Y\leq y) = \mathbb P(X\leq x)\mathbb P(Y\leq y)$, for all $x,y\in\mathbb R$, and so on. (a) does not imply independence in general. (c) ($\mathbb P(X\leq x,Y\leq y) \geq \mathbb P(X\leq x)\mathbb P(Y\leq y)$) is often refereed to that $X$ and $Y$ are positively associated.2011-01-30

3 Answers 3