In probability theory, a random variable Y {\displaystyle Y} is said to be mean independent of random variable X {\displaystyle X} if and only if its conditional mean E ( Y ∣ X = x ) {\displaystyle E(Y\mid X=x)} equals its (unconditional) mean E ( Y ) {\displaystyle E(Y)} for all x {\displaystyle x} such that the probability density/mass of X {\displaystyle X} at x {\displaystyle x} , f X ( x ) {\displaystyle f_{X}(x)} , is not zero. Otherwise, Y {\displaystyle Y} is said to be mean dependent on X {\displaystyle X} . Stochastic independence implies mean independence, but the converse is not true.; 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 {\displaystyle Y} to be mean-independent of X {\displaystyle X} even though X {\displaystyle X} is mean-dependent on Y {\displaystyle Y} . The concept of mean independence is often used in econometrics to have a middle ground between the strong assumption of independent random variables ( X 1 ⊥ X 2 {\displaystyle X_{1}\perp X_{2}} ) and the weak assumption of uncorrelated random variables ( Cov ( X 1 , X 2 ) = 0 ) . {\displaystyle (\operatorname {Cov} (X_{1},X_{2})=0).}
Further reading Cameron, A. Colin; Trivedi, Pravin K. (2009). Microeconometrics: Methods and Applications (8th ed.). New York: Cambridge University Press. ISBN 9780521848053. Wooldridge, Jeffrey M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). London: The MIT Press. ISBN 9780262232586.
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