# Correlated/dependent random variables

This may be a very basic question or a very difficult one, I am not sure.

What does it mean if two random variables are perfectly correlated? What does it mean if two random variables are perfectly dependent (whatever that means)?

When two random variables are independent joint pdf can be factorized into marginal pdfs. But if they are say perfectly correlated, what does that say? Does it mean they come from distributions with same functional forms but with different variances?

What does it mean to be perfectly dependent? Conditional distribution collapses to a dirac delta function?

Thanks

• Out of curiosity, where have you seen the phrase "perfectly dependent"? Do you have a reference or link?
– whuber
Commented Aug 3, 2013 at 18:42
• Nowhere really. "Perfectly correlated" abundant however. Correlation is one measure of dependency. That's why probably makes more sense in that context. It may make sense in other measures of dependency as well I suppose, but not the dependency itself. Commented Aug 3, 2013 at 18:50
• That's good to know, because according to the standard definition of dependence in statistics, "perfectly dependent" is nonsensical :-).
– whuber
Commented Aug 3, 2013 at 20:00

I think one could interpret "two random variables are perfectly correlated" as "the correlation of the two random variables is $1$ (or maybe $\pm 1$)", and interpret "the two random variables are perfectly dependent" as "two random variables are comonotonic (or maybe comonotonic or countermonotonic)"

Perfect linear dependence

Let $X$ and $Y$ be two random variables. If $|\mbox{corr}(X,Y)| = 1$, then $Y = \alpha + \beta X$ almost surely for some $\alpha \in \mathbb{R}$ and $\beta \neq 0$, with $\beta > 0$ for positive linear dependence and $\beta < 0$ for negative linear dependence.

It can be shown that $\mbox{corr}(X,Y) = 1$ if and only if $X$ and $Y$ are of the same type, that is there exists constants $a\in \mathbb{R}$ and $b > 0$ such that $Y \stackrel{\mathrm{d}}{=} a + bY$, where $\stackrel{\mathrm{d}}{=}$ denote equallity in distribution, meaning that $X$ and $Y$ have the same distribution up to a location and scale transformation.

Perfect positive dependence

The random variables $X$ and $Y$ are comonotonic if they admit as copula the Fréchet upper bound, that is the copula associated to $X$ and $Y$ is $C(u,v) = \min(u,v)$, which is the strongest type of "positive" dependence. It can be shown that $X$ and $Y$ are comonotonic if and only if $$(X,Y) \stackrel{\mathrm{d}}{=} (h_1(Z), h_2(Z))$$ for some random variable $Z$ and increasing functions $h_1$ and $h_2$. So, comonotonic random variables are only functions of a single random variable; the conditional density of $Y$ given $X = x$ is a Dirac delta function.

The joint distribution of $X$ and $Y$ is $$F(x,y) = C(F_X(x), F_Y(y)) = \min(F_X(x), F_Y(y)) ,$$ where $F_X$ and $F_Y$ denote the marginal distributions of $X$ and $Y$.

• Excellent take; I like the way you started with the interpretation (which I'd say is likely the best way t take the question) and then wrote a clear answer. Commented Apr 28, 2014 at 8:01
• In your answer for Perfect Linear Dependence, do you mean $Y = a + bX$? Commented Jul 6, 2015 at 2:19