Mean dependence

In probability theory, a random variable $Y$ is said to be mean independent of random variable $X$ if and only if its conditional mean $E(Y\; \backslash mid\; X\; =\; x)$ equals its (unconditional) mean $E(Y)$ for all $x$ such that the probability density/mass of $X$ at $x$, $f\_X(x)$, is not zero. Otherwise, $Y$ is said to be mean dependent on $X$.

Stochastic independence implies mean independence, but the converse is not true.;{{Harvtxt|Cameron|Trivedi|2009|p=23}}{{Harvtxt|Wooldridge|2010|pp=54, 907}} moreover, mean independence implies uncorrelatedness while the converse is not true. Unlike stochastic independence and uncorrelatedness, mean independence is not symmetric: it is possible for $Y$ to be mean-independent of $X$ even though $X$ is mean-dependent on $Y$.

The concept of mean independence is often used in econometrics{{citation needed|date=February 2016}} to have a middle ground between the strong assumption of independent random variables ($X\_1\; \backslash perp\; X\_2$) and the weak assumption of uncorrelated random variables $(\backslash operatorname\{Cov\}(X\_1,\; X\_2)\; =\; 0).$